Perron similarities and the nonnegative inverse eigenvalue problem

Abstract.

The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known.

An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella [Linear Algebra Appl., 493 (2016), 281–300] developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities.

Each Perron similarity gives a nontrivial polyhedral cone and convex polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevič arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of four-by-four stochastic matrices.

Similar to the Karpelevič region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. Extremal elements of the set are defined and shown to be on the boundary.

It is shown that the polyhedral cone and convex polytope of the discrete Fourier transform (DFT) matrix corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices.

Implications for further inquiry are also given.

2020 Mathematics Subject Classification:

Primary: 15A29; Secondary: 15B48, 15B51

1. Introduction

The nonnegative inverse eigenvalue problem (NIEP) asks, for each positive integer $n$ , which multi-sets $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ of $n$ complex numbers occur as the eigenvalues of an $n$ -by- $n$ entry-wise nonnegative matrix? This has proven one of the most challenging problems in mathematics, and is certainly the most sought-after question in matrix analysis. Thus, a variety of sub-questions have been worthy goals.

If $A\geqslant 0$ (entry-wise) is $n$ -by- $n$ and with spectrum $\Lambda$ , then $\Lambda$ is called realizable, and $A$ is called a realizing matrix. If the realizing matrix is required to be diagonalizable, then the resulting subproblem is called the diagonalizable NIEP or the DNIEP. There are differences between the two problems [16, p. 214] and both are unsolved when $n>4$ . A solution to either has appeared far off. For additional information about the NIEP, and its numerous variants, there is a recent survey [16].

It is known that if $\Lambda=\{\lambda_{1},\dots,\lambda_{n}\}$ is realizable and $A$ is a realizing matrix for $\Lambda$ , then

(1.1)

\rho\left(\Lambda\right)\coloneqq\max_{1\leqslant k\leqslant n}\left\{|\lambda_{k}|\right\}\in\Lambda,

(1.2)

\Lambda=\overline{\Lambda}\coloneqq\left\{\overline{\lambda_{1}},\dots,\overline{\lambda_{n}}\right\},

(1.3)

s_{k}(\Lambda)\coloneqq\sum_{i=1}^{n}\lambda_{i}^{k}=\tr{(A^{k})}\geqslant 0,~{}\forall k\in\mathbb{N},

and

(1.4)

\left[s_{k}(\Lambda)\right]^{\ell}\leqslant n^{\ell-1}s_{k\ell}(\Lambda),\forall k,\ell\in\mathbb{N}.

These conditions are not independent: Loewy and London [25] showed that the moment condition (1.3) implies the self-conjugacy condition (1.2). Friedland [9, Theorem 1] showed that the eventual nonnegativity of the moments implies the spectral radius condition (1.1). Finally, if the trace is nonnegative, i.e., if $s_{1}(\Lambda)\geqslant 0$ , then the JLL condition (1.4) (established independently by Johnson [15] and by Loewy and London [25]) implies the moment condition since

s_{\ell}(\Lambda)\geqslant\frac{1}{n^{\ell-1}}\left[s_{1}(\Lambda)\right]^{\ell}\geqslant 0,\forall\ell\in\mathbb{N}.

Thus, the JLL condition and the nonnegativity of the trace imply the self-conjugacy, spectral radius, and moment conditions.

Holtz [11] showed that if $\Lambda=\{\lambda_{1},\dots,\lambda_{n}\}$ is realizable, with $\lambda_{1}=\rho\left(\Lambda\right)$ , then the shifted spectrum $\{0,\lambda_{1}-\lambda_{2},\dots,\lambda_{1}-\lambda_{n}\}$ satisfies Newton’s inequalities. Furthermore, Holtz demonstrated that these inequalities are independent of (1.3) an (1.4).

The problem of characterizing the nonzero spectra of nonnegative matrices is due to Boyle and Handelman [3] (a constructive version of their main result was given by Laffey [23]). However, despite their remarkable achievement, and the stringent necessary conditions listed above, the NIEP and its variants are unsolved when $n$ is greater than four.

Our focus here is upon the DNIEP (without loss of generality, when the realizing matrix is irreducible). We are able to explicitly characterize invertible matrices that diagonalize irreducible nonnegative matrices. We call them Perron similarities. We show also that, for each Perron similarity, there is a nontrivial polyhedral cone of realizable spectra, which we call the (Perron) spectracone. When the Perron similarity is properly normalized, the cross section of the spectracone, for which the spectral radius is one, is called the (Perron) spectratope and is a convex polytope. With the mentioned normalization, each matrix may be taken to be row stochastic. This focuses attention, for each Perron similarity, upon the extreme points that are finite in number. Their determination solves a dramatic portion of the NIEP. This program is carried out for particular types of matrices, such as those that correspond to Type I Karpelevič arcs (see below) and circulant and block circulant matrices.

2. Notation & Background

For ease of notation, $\mathbb{N}$ denotes the set of positive integers and $\mathbb{N}_{0}\coloneqq\mathbb{N}\cup\{0\}$ . If $n\in\mathbb{N}$ , then $\left[n\right]\coloneqq\{k\in\mathbb{N}\mid 1\leqslant k\leqslant n\}$ .

The set of $m$ -by- $n$ matrices with entries from a field $\mathbb{F}$ is denoted by $\mathsf{M}_{m\times n}(\mathbb{F})$ . If $m=n$ , then $\mathsf{M}_{n\times n}(\mathbb{F})$ is abbreviated to $\mathsf{M}_{n}(\mathbb{F})$ . The set of nonsingular matrices in $\mathsf{M}_{n}(\mathbb{F})$ is denoted by $\mathsf{GL}_{n}\left(\mathbb{F}\right)$ .

If $x\in\mathbb{F}^{n}$ , then $x_{k}$ or $[x]_{k}$ denotes the $k\textsuperscript{th}$ -entry of $x$ and $D_{x}=D_{x^{\top}}\in\mathsf{M}_{n}$ denotes the diagonal matrix whose $(i,i)$ -entry is $x_{i}$ . Notice that

D_{\alpha x+\beta y}=\alpha D_{x}+\beta D_{y},\ \forall\alpha,\beta\in\mathbb{F},\forall x,y\in\mathbb{F}^{n}.

Denote by $I$ , $e$ , $e_{i}$ , and $0$ the identity matrix, the all-ones vector, the $i\textsuperscript{th}$ canonical basis vector, and the zero vector, respectively. The size of each aforementioned object is implied by its context.

If $A\in\mathsf{M}_{m\times n}(\mathbb{F})$ , then:

•

$a_{ij}$ , $a_{i,j}$ , or $[A]_{ij}$ denotes the $(i,j)$ -entry of $A$ ;
•

$A^{\top}$ denotes the transpose of $A$ ;
•

$\overline{A}=[\overline{a_{ij}}]$ denotes the entrywise conjugate of $A$ ;
•

$A^{\ast}\coloneqq\overline{A^{\top}}=\overline{A}^{\top}$ denotes the conjugate-transpose of $A$ ; and
•

$r_{i}(A)\coloneqq A^{\top}e_{i}$ denotes the $i\textsuperscript{th}$ -row of $A$ as a column vector (when the context is clear, $r_{i}(A)$ is abbreviated to $r_{i}$ ).

If $A\in\mathsf{M}_{n}(\mathbb{F})$ , then $\operatorname{spec}A=\operatorname{spec}(A)$ denotes the spectrum of $A$ and $\rho=\rho(A)$ denotes the spectral radius of $A$ .

If $A\in\mathsf{M}_{n}(\mathbb{F})$ and $n\geqslant 2$ , then $A$ is called reducible if there is a permutation matrix $P$ such that

\displaystyle P^{\top}AP=\begin{bmatrix}A_{11}&A_{12}\\ 0&A_{22}\end{bmatrix},

where $A_{11}$ and $A_{22}$ are nonempty square matrices. If $A$ is not reducible, then A is called irreducible.

If $A\in\mathsf{M}_{n}(\mathbb{F})$ , then the characteristic polynomial of $A$ , denoted by $\chi_{A}$ , is defined by $\chi_{A}(t)=\det(tI-A)$ . The companion matrix $C=C_{p}$ of a monic polynomial $p(t)=t^{n}+\sum_{k=1}^{n}c_{k}t^{n-k}$ is the $n$ -by- $n$ matrix defined by

C=\left[\begin{array}[]{cc}0&I_{n-1}\\ -c_{n}&-c\end{array}\right],

where $c=[c_{n-1}~{}\cdots~{}c_{1}]$ . It is well-known that $\chi_{C_{p}}=p$ . Notice that $C$ is irreducible if and only if $c_{n}\neq 0$ .

If $x,y\in\mathbb{C}^{n}$ , then $\langle x,y\rangle$ denotes the canonical inner product of $x$ and $y$ , i.e.,

\langle x,y\rangle=y^{\ast}x=\sum_{k=1}^{n}\overline{y_{k}}\cdot x_{k}

and

\begin{Vmatrix}x\end{Vmatrix}_{2}\coloneqq\sqrt{\langle x,x\rangle}.

If $n\in\mathbb{N}$ , then

S^{n}\coloneqq\left\{x\in\mathbb{C}^{n}\mid||x||_{\infty}\coloneqq\max_{1\leqslant k\leqslant n}\{|x_{k}|\}=1\right\}

and $B^{n}\coloneqq\{x\in\mathbb{C}^{n}\mid||x||_{\infty}\leqslant 1\}$ .

The Hadamard product of $A=[a_{ij}]$ , $B=[b_{ij}]\in\mathsf{M}_{m\times n}(\mathbb{F})$ , denoted by $A\circ B$ , is the $m$ -by- $n$ matrix whose $(i,j)$ -entry is $a_{ij}b_{ij}$ . If $x\in\mathbb{F}^{n}$ and $p\in\mathbb{N}$ , then $x^{p}$ denotes the $p\textsuperscript{th}$ -power of $x$ with respect to the Hadamard product, i.e., $[x^{p}]_{k}=x_{k}^{p}$ . If $p=0$ , then $x^{p}\coloneqq e$ . If $x\in\mathbb{F}^{n}$ is totally nonzero, i.e., $x_{k}\neq 0,\ \forall k\in\left[n\right]$ , then $x^{-1}$ denotes the inverse of $x$ with respect to the Hadamard product, i.e., $[x^{-1}]_{k}=x_{k}^{-1}$ . Notice that if $x$ is totally nonzero, then $(D_{x})^{-1}=D_{x^{-1}}$ .

The direct sum of $A_{1},\dots,A_{\ell}$ , where $A_{k}\in\mathsf{M}_{n_{k}}(\mathbb{F})$ , denoted by $A_{1}\oplus\dots\oplus A_{\ell}$ , or $\bigoplus_{k=1}^{\ell}A_{k}$ , is the $n$ -by- $n$ matrix

\left[\begin{array}[]{ccc}A_{1}&&\hbox{\multirowsetup\Large 0}\\ \hbox{\multirowsetup\Large 0}&\ddots&\\ &&A_{\ell}\end{array}\right],\ n=\sum_{k=1}^{\ell}n_{k}.

If $\sigma\in\mathsf{Sym}(n)$ and $x\in\mathbb{C}^{n}$ , then $\sigma(x)$ is the $n$ -by- $1$ vector such that $[\sigma(x)]_{k}=x_{\sigma(k)}$ and $P_{\sigma}\in\mathsf{M}_{n}$ denotes the permutation matrix corresponding to $\sigma$ , i.e., $P_{\sigma}$ is the the $n$ -by- $n$ matrix whose $(i,j)$ -entry is $\delta_{\sigma(i),j}$ , where $\delta_{ij}$ denotes the Kronecker delta. As is well-known, $\left(P_{\sigma}\right)^{-1}=P_{\sigma^{-1}}=\left(P_{\sigma}\right)^{\top}$ . When the context is clear, $P_{\sigma}$ is abbreviated to $P$ . Notice that $Px=\sigma(x)$ .

If $k\in\left[n\right]$ , then $P_{k}$ denotes the matrix obtained by deleting the $k\textsuperscript{th}$ -row of $I_{n}$ and $\pi_{k}:\mathbb{F}^{n}\longrightarrow\mathbb{F}^{n-1}$ is the projection map defined by $\pi_{k}(x)=P_{k}x$ .

2.1. Nonnegative Matrices

If $A\in\mathsf{M}_{n}(\mathbb{R})$ and $a_{ij}\geqslant 0,\ \forall i,j\in\left[n\right]$ or $a_{ij}>0,\ \forall i,j\in\left[n\right]$ , then $A$ is called nonnegative or positive, respectively, and we write $A\geqslant 0$ or $A>0$ , respectively. If $x\in\mathbb{C}^{n}$ ( $A\in\mathsf{M}_{n}(\mathbb{C})$ ), then $x\geqslant 0$ (respectively, $A\geqslant 0$ ) if $\real x\geqslant 0$ and $\imaginary x=0$ (respectively, $\real A\geqslant 0$ and $\imaginary A=0$ ).

If $A\geqslant 0$ and

\sum_{j=1}^{n}a_{ij}=1,~{}\forall~{}i\in\left[n\right],

then $A$ is called (row) stochastic. If $A\geqslant 0$ , then $A$ is stochastic if and only if $Ae=e$ . Furthermore, if $A$ is stochastic, then $1\in\operatorname{spec}(A)$ and $\rho(A)=1$ . It is known that the NIEP and the stochastic NIEP are equivalent (see, e.g., Johnson [15, p. 114]).

Observation 2.1.

\thlabel

stochasticconvex If $A_{1},\ldots,A_{m}\in\mathsf{M}_{n}(\mathbb{R})$ are stochastic, then the matrix

A\coloneqq\sum_{k=1}^{m}\alpha_{k}A_{k},\ \sum_{k=1}^{m}\alpha_{k}=1,\ \alpha_{k}\geqslant 0,\ \forall k\in\left[m\right],

is stochastic.

We recall the Perron–Frobenius theorem for irreducible matrices.

Theorem 2.2 ([13, Theorem 8.4.4]).

\thlabel

pftirr Let $A\in\mathsf{M}_{n}(\mathbb{R})$ be irreducible and nonnegative, and suppose that $n\geqslant 2$ . Then

(i)

$\rho(A)>0$ ;
(ii)

$\rho(A)$ is an algebraically simple eigenvalue of $A$ ;
(iii)

there is a unique positive vector $x$ such that $Ax=\rho(A)x$ and $e^{\top}x=1$ ; and
(iv)

there is a unique positive vector $y$ such that $y^{\top}A=\rho(A)y^{\top}$ and $e^{\top}y=1$ .

The vector $x$ in part (iii) of \threfpftirr is called the (right) Perron vector of $A$ and the vector $y$ in part (iv) of \threfpftirr is called the left Perron vector of A [13].

3. Preliminary Results

In this section, we cover more background and establish some ancillary results that will be useful in the sequel.

3.1. Complex Polyhedral Regions

If $S\subseteq\mathbb{C}^{n}$ , then the conical hull of $S$ , denoted by $\operatorname{coni}S=\operatorname{coni}(S)$ , is defined by

\operatorname{coni}{S}=\begin{cases}\{0\},&S=\varnothing\\ \left\{\sum_{k=1}^{m}\alpha_{k}x_{k}\in\mathbb{C}^{n}\mid m\in\mathbb{N},~{}x_{k}\in S,~{}\alpha_{k}\geqslant 0\right\},&S\neq\varnothing.\end{cases}

i.e., when $S$ is nonempty, $\operatorname{coni}{S}$ consists of all conical combinations. Similarly, the convex hull of $S$ , denoted by $\operatorname{conv}{S}=\operatorname{conv}(S)$ is defined by

\operatorname{coni}{S}=\begin{cases}\{0\},&S=\varnothing\\ \left\{\sum_{k=1}^{m}\alpha_{k}x_{k}\in\mathbb{C}^{n}\mid m\in\mathbb{N},~{}x_{k}\in S,\ \sum_{k=1}^{m}\alpha_{k}=1,\ \alpha_{k}\geqslant 0\right\},&S\neq\varnothing\end{cases}.

The conical hull (convex hull) of a finite list $\{x_{1},\ldots,x_{n}\}$ is abbreviated to $\operatorname{coni}(x_{1},\ldots,x_{n})$ (respectively, $\operatorname{conv}(x_{1},\ldots,x_{n})$ ).

A subset set $K$ of $\mathbb{C}^{n}$ is called

•

a cone if $\operatorname{coni}x\subseteq K,\ \forall x\in K$ ;
•

convex if $\operatorname{conv}(x,y)\subseteq K,\ \forall x,y\in K$ ;
•

star-shaped at $c\in K$ if $\operatorname{conv}(c,x)\subseteq K,\ \forall x\in K$ ; and
•

a convex cone if $\operatorname{coni}(x,y)\subseteq K,\ \forall x,y\in K$ .

If $S,\ T\subseteq\mathbb{C}^{n}$ , then $S+T$ denotes the Minkowski sum of $S$ and $T$ , i.e., $S+T\coloneqq\{x\in\mathbb{C}^{n}\mid x=s+t,\ s\in S,\ t\in T\}$ and $-S\coloneqq\{x\in\mathbb{C}^{n}\mid x=-s,\ s\in S\}$ . If $K$ is a convex cone, then the dimension of $K$ is the quantity $\dim(K+(-K))$ . If $K=\operatorname{coni}(x_{1},\ldots,x_{k})$ , then $\dim K=\dim(\operatorname{span}(x_{1},\ldots,x_{k}))$ .

A point $x$ of a convex cone $K$ is called an extreme direction or extreme ray if $u\notin K$ or $v\notin K$ , whenever $x=\alpha u+(1-\alpha)v$ , with $\alpha\in(0,1)$ and $u,v$ linearly independent. A point $x$ of a convex set $K$ is called an extreme point (of $K$ ) if $u\notin K$ or $v\notin K$ , whenever $x=\alpha u+(1-\alpha)v$ , with $\alpha\in(0,1)$ and $u\neq v$ , i.e., $x$ does not lie in any open line segment contained in $K$ .

If $a\in\mathbb{C}^{n}$ , $a\neq 0$ , and $b\in\mathbb{R}$ , then

\mathsf{H}(a,b)\coloneqq\left\{x\in\mathbb{C}^{n}\mid\real(\langle a,x\rangle)\geqslant b\right\}=\left\{x\in\mathbb{C}^{n}\mid\real(\langle x,a\rangle)\geqslant b\right\}

is a closed half-space determined by the hyperplane $\left\{x\in\mathbb{C}^{n}\mid\real(\langle x,a\rangle)=b\right\}$ . If $b=0$ , then the half-space $\mathsf{H}(a,b)$ is abbreviated to $\mathsf{H}(a)$ and contains the origin on its boundary.

Any set of the form

\bigcap_{k=1}^{m}\mathsf{H}(a_{k},b_{k})=\bigcap_{k=1}^{m}\left\{x\in\mathbb{C}^{n}\mid\real(\langle a_{k},x\rangle)\geqslant b_{k}\right\}

is called a polyhedron and a bounded polyhedron is called a polytope. Any set of the form

\bigcap_{k=1}^{m}\mathsf{H}(a_{k})=\left\{x\in\mathbb{C}^{n}\mid\real(\langle a_{k},x\rangle)\geqslant 0\right\}

is called a polyhedral cone.

Since $\langle\langle x,y\rangle\rangle\coloneqq\real(\langle x,a\rangle)$ is a real inner product and $\mathbb{C}^{n}$ is a $2n$ -dimensional real vector space, it follows that $\mathbb{C}^{n}$ is a $2n$ -dimensional Euclidean space and the following celebrated result is applicable (see, e.g., [22, Corollaries 2.13 and 2.14] and [28, pp. 170–178]; or references in [1, Remark 3.3] or [30, p. 87]).

Theorem 3.1 (Farkas–Minkowski–Weyl).

If $P$ is a polytope or a polyhedral cone in a Euclidean space $\mathbb{E}^{n}$ , then there are vectors $x_{1},\ldots,x_{k}$ such that

P=\operatorname{conv}(x_{1},\ldots,x_{k})

P=\operatorname{coni}(x_{1},\ldots,x_{k}),

respectively.

The following four propositions will be useful in the sequel; because they are easy to establish, their proofs are omitted.

Proposition 3.2.

\thlabel

exptball If $n\in\mathbb{N}$ , then $e$ is an extreme point of the convex set $B^{n}$ .

If $n\in\mathbb{N}$ , then the standard (or unit) $n$ -simplex, denoted by $\Delta^{n}$ , is defined by

\Delta^{n}=\left\{(\alpha_{1},\dots,\alpha_{n+1})\in\mathbb{R}^{n+1}\mid\sum_{i=1}^{n+1}\alpha_{i}=1,~{}\alpha_{i}\geqslant 0\right\}.

Proposition 3.3.

\thlabel

exptconv Let $K\subseteq\mathbb{C}^{n}$ be convex and suppose that $x$ is an extreme point of $K$ . If there are vectors $x_{1},\dots,x_{m}\in K$ and a vector $\alpha=[\alpha_{1}~{}\cdots~{}\alpha_{m}]^{\top}\in\Delta^{m-1}$ such that $x=\sum_{k=1}^{m}\alpha_{k}x_{k}$ , then $\alpha\in\{e_{1},\dots,e_{m}\}$ .

Proposition 3.4.

\thlabel

cor:allonesextreme If $S=\{x_{1},\dots,x_{k}\}\subseteq B^{n}$ , then $e\in\operatorname{conv}{(S)}$ if and only if $e\in S$ .

Proposition 3.5.

\thlabel

conisub If $S$ is a finite subset of a convex cone $K$ , then $\operatorname{coni}(S)\subseteq K$ .

Proposition 3.6.

\thlabel

conecontain Let $K$ and $\hat{K}$ be convex cones. If $K=\operatorname{coni}(x_{1},\dots,x_{m})$ , then $K\subseteq\hat{K}$ if and only if $x_{1},\dots,x_{m}\in\hat{K}$ .

3.2. Region Comprising Stochastic Lists

For $x\in\mathbb{C}^{n}$ , denote by $\Lambda(x)$ the list $\{x_{1},\dots,x_{n}\}$ and for every natural number $n$ , let

\mathbb{L}^{n}\coloneqq\{x\in\mathbb{C}^{n}\mid\Lambda(x)=\operatorname{spec}(A),~{}A\in\mathsf{M}_{n}\left(\mathbb{R}\right),~{}A\geqslant 0\}

and

\mathbb{SL}^{n}\coloneqq\{x\in\mathbb{C}^{n}\mid\Lambda(x)=\operatorname{spec}(A),~{}A\in\mathsf{M}_{n}\left(\mathbb{R}\right),~{}A~{}\text{stochastic}\}.

Clearly, $\mathbb{L}^{n}$ is a cone that contains $\mathbb{SL}^{n}$ and a characterization of either set constitutes a solution to the NIEP. As such, we catalog various properties of each set.

Recall that if $x,y\in\mathbb{C}^{n}$ , then the angle $\theta=\theta(x,y)$ between them is defined by

(3.1)

\theta=\theta(x,y)=\begin{cases}\frac{\pi}{2},&(x=0)\vee(y=0)\\ \arccos\frac{\real(\langle x,y\rangle)}{||x||_{2}\cdot||y||_{2}},&(x\neq 0)\wedge(y\neq 0)\end{cases}.

Because non-real eigenvalues of real matrices occur in complex conjugate pairs, it follows that $x\in\mathbb{L}^{n}\iff\bar{x}\in\mathbb{L}^{n}$ . Thus, $\Lambda(x)=\overline{\Lambda(x)}=\Lambda(\bar{x})$ whenever $x\in\mathbb{L}^{n}$ or $\bar{x}\in\mathbb{L}^{n}$ . Furthermore, if $x\in\mathbb{L}^{n}$ , then there is a nonnegative matrix $A$ such that $\operatorname{spec}A=\Lambda(x)=\Lambda(\bar{x})$ . Consequently,

(3.2)

\langle x,e\rangle=e^{\ast}x=e^{\top}x=\sum_{k=1}^{n}x_{k}=\tr{A}\geqslant 0

and

(3.3)

\langle\overline{x},e\rangle=e^{\top}\bar{x}=\sum_{k=1}^{n}\overline{x_{k}}=\tr{A}\geqslant 0

since the realizing matrix is nonnegative.

Proposition 3.7.

\thlabel

SLangle If $x\in\mathbb{L}^{n}$ , then $\theta(x,e)\in[0,\pi/2]$ and $\theta(\bar{x},e)\in[0,\pi/2]$ .

Proof.

Immediate from (3.1), (3.2), and (3.3). ∎

Lemma 3.8.

\thlabel

seq If $\{A_{k}\}_{k=1}^{\infty}$ is a convergent sequence of stochastic $n$ -by- $n$ matrices with limit $L$ , then $L$ is stochastic.

Proof.

Routine analysis exercise. ∎

Theorem 3.9.

If $n$ is a positive integer, then $\mathbb{SL}^{n}$ is compact.

Proof.

Following \threfseq and the fact that the eigenvalues of a matrix are topologically continuous [24, pp. 620–621], it follows that $\mathbb{SL}^{n}$ is closed. Because the spectral radius of a stochastic matrix is one, we obtain $\mathbb{SL}^{n}\subseteq S^{n}$ , i.e., $\mathbb{SL}^{n}$ is bounded. ∎

Theorem 3.10.

If $n$ is a positive integer, then $\mathbb{SL}^{n}$ is star-shaped at $e$ .

Proof.

If $x\in\mathbb{SL}^{n}$ , then there is a stochastic matrix $A$ such that $\Lambda(x)=\operatorname{spec}(A)$ . Write $\operatorname{spec}(A)=\{1,x_{2},\ldots,x_{n}\}$ and let $S$ be an invertible matrix such that $J=S^{-1}AS$ is a Jordan canonical form of $A$ . If $\alpha\in[0,1]$ and $\beta\coloneqq 1-\alpha$ , then the matrix $\alpha A+\beta I$ is stochastic (\threfstochasticconvex) and since $S^{-1}\left(\alpha A+\beta I\right)S=\alpha J+\beta I$ is upper-triangular, it follows that $\operatorname{spec}(\alpha A+\beta I)=\{1,\alpha x_{2}+\beta,\ldots,\alpha x_{n}+\beta\}$ , i.e., $\alpha x+\beta e\in\mathbb{SL}^{n}$ , $\forall\alpha\in[0,1]$ . ∎

If $x\in\mathbb{SL}^{n}$ , then $1=\rho\left(\Lambda(x)\right)\in\Lambda(x)$ and if $P$ is a permutation matrix corresponding to the permutation $\sigma$ , then $\Lambda(Px)=\Lambda(\sigma(x))=\Lambda(x)$ . In light of these two facts, there is no loss in generality in restricting attention to $\mathbb{SL}_{1}^{n}\coloneqq\left\{x\in\mathbb{SL}^{n}\mid x_{1}=1\right\}$ .

The following eigenvalue-perturbation result is due to Brauer [4, Theorem 27] (for more proofs, see [26] and references therein).

Theorem 3.11 (Brauer).

\thlabel

Brauer Let $A\in\mathsf{M}_{n}(\mathbb{C})$ and suppose that

\operatorname{spec}(A)=\{\lambda_{1},\ldots,\lambda_{k},\dots,\lambda_{n}\}.

If $x$ is an eigenvector associated with $\lambda_{k}$ and $y\in\mathbb{C}^{n}$ , then $\operatorname{spec}(A+xy^{*})=\{\lambda_{1},\ldots,\lambda_{k}+y^{*}x,\ldots,\lambda_{n}\}$ .

Theorem 3.12.

If $n>1$ , then $\pi_{1}(\mathbb{SL}_{1}^{n})$ is star-shaped at the origin.

Proof.

If $x\in\mathbb{SL}_{n}$ , then there is a stochastic matrix $A$ such that $\Lambda(x)=\operatorname{spec}A$ . If $\operatorname{spec}A=\{1,x_{2},\ldots,x_{n}\}$ and $\alpha\in\mathbb{C}$ , then $\operatorname{spec}(\alpha A)=\{\alpha,\alpha x_{2},\ldots,\alpha x_{n}\}$ . By \threfstochasticconvex, the matrix

\alpha A+\frac{1-\alpha}{n}ee^{\top}

is stochastic. Since $(\alpha A)e=\alpha e$ , it follows that

	$\displaystyle\operatorname{spec}\left(\alpha A+\frac{1-\alpha}{n}ee^{\top}\right)$	$\displaystyle=\operatorname{spec}\left(\alpha A+e\left(\frac{1-\alpha}{n}e^{\top}\right)\right)$
(\threfBrauer)			$\displaystyle=\left\{\alpha+\frac{1-\alpha}{n}(e^{\top}e),\alpha x_{2},\ldots,\alpha x_{n}\right\}$
		$\displaystyle=\{\alpha+(1-\alpha),\alpha x_{2},\ldots,\alpha x_{n}\}$
		$\displaystyle=\{1,\alpha x_{2},\ldots,\alpha x_{n}\}.$

Thus, $\alpha\pi_{1}(x)\in\pi_{1}(\mathbb{SL}^{n}),\forall\alpha\in[0,1]$ . ∎

3.3. The Karpelevič Region

In 1938, Kolmogorov posed the question of characterizing the region in the complex plane, denoted by $\Theta_{n}$ , that occur as an eigenvalue of a stochastic matrix [32, p. 2]. Dmitriev and Dynkin [6] (see [32, Appendix A] or [31] for an English translation) obtained a partial solution, and Karpelevič [21] (see [31] for an English translation) solved the problem by showing that the boundary of $\Theta_{n}$ consists of curvilinear arcs (hereinafter, Karpelevič arcs or K-arcs), whose points satisfy a polynomial equation that depends on the endpoints of the arc.

If $n\in\mathbb{N}$ , then $\mathcal{F}_{n}\coloneqq\{p/q\mid 0\leq p\leqslant q\leq n,~{}\gcd(p,q)=1\}$ denotes the set of Farey fractions. The following result is the celebrated Karpelevič theorem in a form due to Ito [14].

Theorem 3.13 (Karpelevič theorem).

\thlabel

karpito The region $\Theta_{n}$ is symmetric with respect to the real axis, is included in the unit-disc $\{z\in\mathbb{C}\mid|z|\leq 1\}$ , and intersects the unit-circle $\{z\in\mathbb{C}\mid|z|=1\}$ at the points $\left\{e^{\frac{2\pi p}{q}\mathsf{i}}\mid p/q\in\mathcal{F}_{n}\right\}$ . The boundary of $\Theta_{n}$ consists of these points and of curvilinear arcs connecting them in circular order.

Let the endpoints of an arc be $e^{\frac{2\pi p}{q}\mathsf{i}}$ and $e^{\frac{2\pi r}{s}\mathsf{i}}$ ( $q\leqslant s$ ). Each of these arcs is given by the following parametric equation:

(3.4)

t^{s}\left(t^{q}-\beta\right)^{\left\lfloor n/q\right\rfloor}=\alpha^{\left\lfloor n/q\right\rfloor}t^{q\left\lfloor n/q\right\rfloor},\ \alpha\in[0,1],~{}\beta\coloneqq 1-\alpha.

Following [18], equation (3.4) is called the Ito equation and the polynomial

(3.5)

f_{\alpha}(t)\coloneqq t^{s}(t^{q}-\beta)^{\left\lfloor n/q\right\rfloor}-\alpha^{\left\lfloor n/q\right\rfloor}t^{q\left\lfloor n/q\right\rfloor},\ \alpha\in[0,1],\ \beta\coloneqq 1-\alpha

is called the Ito polynomial. The Ito polynomials are divided into four types as follows (note that $s\neq q\left\lfloor n/q\right\rfloor$ since $\gcd{(q,s)}=1$ ):

•

If $\left\lfloor n/q\right\rfloor=n$ , then

(3.6) $f_{\alpha}^{\mathsf{0}}(t)=(t-\beta)^{n}-\alpha^{n},~{}\alpha\in[0,1],\ \beta\coloneqq 1-\alpha.$

is called a Type 0 (Ito) polynomial and corresponds to the Farey pair $(0/1,1/n)$ .
•

If $\left\lfloor n/q\right\rfloor=1$ , then

(3.7) $f_{\alpha}^{\mathsf{I}}(t)=t^{s}-\beta t^{s-q}-\alpha,\ \alpha\in[0,1],\ \beta\coloneqq 1-\alpha.$

is called a Type I (Ito) polynomial.

•

If $1<\left\lfloor n/q\right\rfloor<n$ and $s>q\left\lfloor n/q\right\rfloor$ , then

(3.8)

f_{\alpha}^{\mathsf{II}}(t)=(t^{q}-\beta)^{\left\lfloor n/q\right\rfloor}-\alpha^{\left\lfloor n/q\right\rfloor}t^{q\left\lfloor n/q\right\rfloor-s},\ \alpha\in[0,1],\ \beta\coloneqq 1-\alpha.

is called a Type II (Ito) polynomial.

•

If $1<\left\lfloor n/q\right\rfloor<n$ and $s<q\left\lfloor n/q\right\rfloor$ , then

(3.9)

f_{\alpha}^{\mathsf{III}}(t)=t^{s-q\left\lfloor n/q\right\rfloor}(t^{q}-\beta)^{\left\lfloor n/q\right\rfloor}-\alpha^{\left\lfloor n/q\right\rfloor},\ \alpha\in[0,1],\ \beta\coloneqq 1-\alpha.

is called a Type III (Ito) polynomial.

The polynomials given by equations (3.6)–(3.9) are called the reduced Ito polynomials. Johnson and Paparella [18, Theorem 3.2] showed that if $\alpha\in[0,1]$ , then there is a stochastic matrix $M=M(\alpha)$ such that $\chi_{M}=f_{\alpha}^{\mathsf{X}}$ , where $\mathsf{X}\in\{\mathsf{0},\mathsf{I},\mathsf{II},\mathsf{III}\}$ .

Remark 3.14.

If $p$ is a polynomial and $p^{\prime}$ denotes its derivative, then $p$ has a multiple root if and only if the resultant $R(p,p^{\prime})\coloneqq\det(S(p,p^{\prime})^{\top})$ vanishes (here $S(p,p^{\prime})$ denotes the Sylvester matrix). Since the coefficients of the polynomials $f_{\alpha}$ and $f_{\alpha}^{\prime}$ depend on the single parameter $\alpha$ , it follows that $\pi(\alpha)\coloneqq\det(S(f_{\alpha},f_{\alpha}^{\prime})^{\top})$ is a univariate polynomial in $\alpha$ of degree at most

\deg f_{\alpha}+(\deg f_{\alpha}-1)=2\deg f_{\alpha}-1.

Hence, there are at most $2\deg f_{\alpha}-1$ zeros and at most $2\deg f_{\alpha}-1$ values in $[0,1]$ such that $f_{\alpha}$ does not have distinct zeros.

More is known about the number of zeros of $f_{\alpha}$ corresponding to the Type I arc $K_{n}({1}/{n},{1}/{(n-1)})$ .

Proposition 3.15.

[17, Proposition 4.1] For $n\geqslant 4$ , let

(3.10)

f_{\alpha}(t)\coloneqq t^{n}-\beta t-\alpha,~{}\alpha\in[0,1],~{}\beta\coloneqq 1-\alpha.

(i)

If $n$ is even, then $f_{\alpha}$ has $n$ distinct roots.
(ii)

If $n$ is odd and $\alpha\geqslant\beta$ , then $f_{\alpha}$ has $n$ distinct roots.
(iii)

If $n$ is odd and $\alpha<\beta$ , then $f_{\alpha}$ has a multiple root if and only if $n^{n}\alpha^{n-1}-(n-1)^{n-1}\beta^{n}=0$ .

If $f_{\alpha}$ is defined as in (3.10), $n$ is odd, $\alpha<\beta$ , and $\pi(\alpha)=(n-1)^{n-1}\beta^{n}-n^{n}\alpha^{n-1}$ , then it is known that the polynomial $\pi$ has only one zero in the interval $[0,1]$ [18, Remark 4.3].

3.4. Extremal and Boundary Points

If $\lambda\in\Theta_{n}$ , then $\lambda$ is called extremal if $\alpha\lambda\not\in\Theta_{n}$ whenever $\alpha>1$ . It is an easy exercise to show that if $\lambda$ is extremal, then $\lambda\in\partial\Theta_{n}$ . Karpelevič asserted that the converse follows from the closure of $\Theta_{n}$ but this is incorrect in view of the two- and three-dimensional cases. However, an elementary proof was given recently by Munger et al. [27, Section 6].

Definition 3.16.

If $x=\begin{bmatrix}1&x_{2}&\cdots&x_{n}\end{bmatrix}^{\top}\in\mathbb{SL}_{1}^{n}$ , then $x$ is called extremal (in $\mathbb{SL}_{1}^{n}$ ) if $\alpha\pi_{1}(x)\not\in\pi_{1}(\mathbb{SL}_{1}^{n})$ , $\forall\alpha>1$ . The set of extremal points in $\mathbb{SL}_{1}^{n}$ is denoted by $\mathbb{E}_{n}$ .

Theorem 3.17.

If $n$ is a positive integer, then $\mathbb{E}_{n}\subseteq\partial\mathbb{SL}_{1}^{n}$ .

Proof.

If $n=1$ , then $\mathbb{SL}_{1}^{n}=\{1\}$ and the result is clear.

Otherwise, assume that $n>1$ and, for contradiction, that $x\in\mathbb{E}_{n}$ , but $x\notin\partial\mathbb{SL}_{1}^{n}$ . By definition, $\exists\varepsilon>0$ such that $N_{\varepsilon}(x)\coloneqq\{y\in\mathbb{C}^{n}\mid\begin{Vmatrix}y-x\end{Vmatrix}_{\infty}<\varepsilon\}\subseteq\mathbb{SL}_{1}^{n}$ . If $\alpha\coloneqq 1+\varepsilon$ and

y\coloneqq\begin{bmatrix}1\\ \pi_{1}(\alpha x)\end{bmatrix}=\begin{bmatrix}1\\ \alpha x_{2}\\ \vdots\\ \alpha x_{n}\end{bmatrix},

then

y-x=\begin{bmatrix}0\\ (\alpha-1)x_{2}\\ \vdots\\ (\alpha-1)x_{n}\end{bmatrix}=\varepsilon\begin{bmatrix}0\\ x_{2}\\ \vdots\\ x_{n}\end{bmatrix}.

Because $\begin{Vmatrix}x\end{Vmatrix}_{\infty}=1$ , it follows that $\begin{Vmatrix}\pi_{1}(x)\end{Vmatrix}_{\infty}\leqslant 1$ and $\begin{Vmatrix}y-x\end{Vmatrix}_{\infty}=\varepsilon\begin{Vmatrix}\pi_{1}(x)\end{Vmatrix}_{\infty}\leqslant\varepsilon$ , i.e., $y\in N_{\varepsilon}(x)$ . Since $\alpha>1$ , it follows that $x$ is not extremal, a contradiction. ∎

Observation 3.18.

If $x\in\mathbb{SL}_{1}^{n}$ and $x_{k}$ is extremal in $\Theta_{n}$ , where $1<k\leqslant n$ , then $x\in\mathbb{E}_{n}$ .

Proof.

For contradiction, if $x\notin\mathbb{E}_{n}$ , then $\exists\alpha>1$ such that $\pi_{1}(x)\in\pi_{1}(\mathbb{SL}_{1}^{n})$ . Thus, there is a stochastic matrix $A$ with spectrum $\{1,\alpha x_{2},\ldots,\alpha_{n}x_{n}\}$ . Consequently, $\alpha x_{k}\in\Theta_{n}$ , a contradiction. ∎

4. Spectral Polyhedra

Although the following results are specified for complex matrices, many of the definitions and results apply, with minimal alteration, to real matrices.

Definition 4.1.

\thlabel

spectratope If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then:

(i)

$\mathcal{C}(S)\coloneqq\{x\in\mathbb{C}^{n}\mid M_{x}\coloneqq SD_{x}S^{-1}\geqslant 0\}$ is called the (Perron) spectracone of $S$ ;
(ii)

$\mathcal{P}(S)\coloneqq\{x\in\mathcal{C}(S)\mid M_{x}e=e\}$ is called the (Perron) spectratope of $S$ ;
(iii)

$\mathcal{A}(S)\coloneqq\{M_{x}\in\mathsf{M}_{n}(\mathbb{R})\mid x\in\mathcal{C}(S)\}$ .

Remark 4.2.

Although the spectratope definitions that appeared in the literature previously ([17, Definition 3.5] and [7, p. 114]) differ from what appears in \threfspectratope, the definition above subsumes the previous definitions and captures the notion in its fullest generality.

Observation 4.3.

\thlabel

prodstochequalsstoch The product of stochastic matrices is stochastic.

Theorem 4.4.

\thlabel

hadamardcones If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then:

(i)

$\mathcal{C}(S)$ is a nonempty convex cone that is closed with respect to the Hadamard product;
(ii)

$\mathcal{P}(S)$ is a nonempty convex set that is closed with respect to the Hadamard product; and
(iii)

$\mathcal{A}(S)$ is a nonempty convex cone that is closed with respect to matrix multiplication.

Proof.

Since $M_{e}=SD_{e}S^{-1}=SI_{n}S^{-1}=I_{n}\geqslant 0$ and $I_{n}$ is stochastic, it follows that $e\in\mathcal{P}(S)\subset\mathcal{C}(S)$ and all three sets are nonempty.

(i)

If $x,y\in\mathcal{C}(S)$ and $\alpha$ , $\beta\geqslant 0$ , then

(4.1)

M_{\alpha x+\beta y}=SD_{\alpha x+\beta y}S^{-1}=S(\alpha D_{x}+\beta D_{y})S^{-1}=\alpha M_{x}+\beta M_{y}\geqslant 0,

i.e., $\mathcal{C}(S)$ is a convex cone.

Furthermore,

(4.2)

M_{x\circ y}=SD_{x\circ y}S^{-1}=SD_{x}D_{y}S^{-1}=(SD_{x}S^{-1})(SD_{y}S^{-1})=M_{x}M_{y}\geqslant 0,

i.e., the convex cone $\mathcal{C}(S)$ is closed with respect to the Hadamard product.

(ii)

The convexity of $\mathcal{P}(S)$ follows from (4.1) and \threfstochasticconvex; closure with respect to the Hadamard product is a consequence of (4.2) and \threfprodstochequalsstoch.
(iii)

Follows from (4.1) and (4.2). ∎

Remark 4.5.

If $\mathcal{C}(S)=\operatorname{coni}(e)$ , $\mathcal{P}(S)=\{e\}$ , or $\mathcal{A}(S)=\operatorname{coni}(I_{n})$ , then $\mathcal{C}(S)$ , $\mathcal{P}(S)$ , and $\mathcal{A}(S)$ are called trivial; otherwise, they are called nontrivial.

Before we prove our next result, we note the following: if $x\in\mathbb{C}^{n}$ , then

\real x\coloneqq\begin{bmatrix}\real x_{1}\\ \vdots\\ \real x_{n}\end{bmatrix}\in\mathbb{R}^{n}

and

\imaginary x\coloneqq\begin{bmatrix}\imaginary x_{1}\\ \vdots\\ \imaginary x_{n}\end{bmatrix}\in\mathbb{R}^{n}.

Since $\mathsf{i}x=-\imaginary x+\mathsf{i}\real x$ and $-\mathsf{i}x=\imaginary x-\mathsf{i}\real x$ , it follows that

(4.3)

\imaginary x=0\iff(\real(\mathsf{i}x)\geqslant 0)\wedge(\real(-\mathsf{i}x)\geqslant 0).

Similarly, if $A\in\mathsf{M}_{n}(\mathbb{C})$ , then $\real A\coloneqq\begin{bmatrix}\real a_{ij}\end{bmatrix}\in\mathsf{M}_{n}(\mathbb{R})$ and $\imaginary A\coloneqq\begin{bmatrix}\imaginary a_{ij}\end{bmatrix}\in\mathsf{M}_{n}(\mathbb{R})$ .

Theorem 4.6.

\thlabel

CSpolyconePSpolytope If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then $\mathcal{C}(S)$ is a polyhedral cone and $\mathcal{P}(S)$ is a polytope.

Proof.

In what follows, we let $t_{ij}$ denote the $(i,j)$ -entry of $S^{-1}$ . Via the mechanics of matrix multiplication and the complex inner product,

(4.4)

\left[M_{x}\right]_{ij}=\left[SD_{x}S^{-1}\right]_{ij}=\sum_{k=1}^{n}\left(s_{ik}t_{kj}\right)x_{k}=\left(s_{i}\circ t_{j}\right)^{\top}x=\langle x,\overline{s_{i}\circ t_{j}}\rangle,

where $s_{i}$ denotes the $i\textsuperscript{th}$ -row of $S$ (as a column vector) and $t_{j}$ denotes the $j\textsuperscript{th}$ -column of $S^{-1}$ . Consequently,

	$\displaystyle x\in\mathcal{C}(S)$	$\displaystyle\iff(\real M_{x}\geqslant 0)\wedge(\imaginary M_{x}=0)$
		$\displaystyle\iff x\in\mathsf{H}\left(\overline{s_{i}\circ t_{j}}\right)\cap\mathsf{H}\left(\mathsf{i}\cdot\overline{s_{i}\circ t_{j}}\right)\cap\mathsf{H}\left(-\mathsf{i}\cdot\overline{s_{i}\circ t_{j}}\right),\ \forall(i,j)\in\left[n\right]^{2}.$

Via the mechanics of matrix multiplication and the complex inner product, notice that

\displaystyle\left[M_{x}e\right]_{i}=\sum_{j=1}^{n}\sum_{k=1}^{n}\left(s_{ik}t_{kj}\right)x_{k}=\sum_{k=1}^{n}\left(s_{ik}\cdot\sum_{j=1}^{n}t_{kj}\right)x_{k}=\left(s_{i}\circ Te\right)^{\top}x=\langle x,\overline{s_{i}\circ Te}\rangle.

Thus, $x\in\mathcal{P}(S)$ if and only if

x\in\mathsf{H}\left(\overline{s_{i}\circ t_{j}}\right)\cap\mathsf{H}\left(\mathsf{i}\cdot\overline{s_{i}\circ t_{j}}\right)\cap\mathsf{H}\left(-\mathsf{i}\cdot\overline{s_{i}\circ t_{j}}\right),\ \forall(i,j)\in\left[n\right]^{2}

and

x\in\mathsf{H}\left(\overline{s_{i}\circ Te},1\right)\cap\mathsf{H}\left(\overline{s_{i}\circ Te},-1\right)\cap\mathsf{H}\left(\mathsf{i}\cdot\overline{s_{i}\circ Te}\right)\cap\mathsf{H}\left(-\mathsf{i}\cdot\overline{s_{i}\circ Te}\right).

Thus, $\mathcal{P}(S)$ is a polyhedron and since $||x||_{\infty}=1$ , it follows that $\mathcal{P}(S)$ is a polytope. ∎

Theorem 4.7.

If $x,y\in\mathcal{C}(S)$ , then $\theta(x,y)\in[0,\pi/2]$ .

Proof.

If $x,y\in\mathcal{C}(S)$ , then, since $\overline{y}\in\mathcal{C}(S)$ , it follows that $x\circ\overline{y}\in\mathcal{C}(S)$ by part (i) of \threfhadamardcones. Since

\langle x,y\rangle=y^{\ast}x=\sum_{k=1}^{n}\overline{y_{k}}\cdot x_{k}=\sum_{k=1}^{n}1\left(x_{k}\cdot\overline{y_{k}}\right)=e^{\top}(x\circ\overline{y})=e^{\ast}(x\circ\overline{y})=\langle x\circ\overline{y},e\rangle,

the result follows from \threfSLangle. ∎

4.1. Basic Transformations

As detailed in the sequel, the set $\mathcal{C}(S)$ is unchanged, or changes predictably, by certain basic transformations of $S$ .

The following result is a routine exercise.

Lemma 4.8.

\thlabel

nonnegsims If $A\in\mathsf{M}_{n}(\mathbb{R})$ , $P$ is a permutation matrix, and $v>0$ , then

(i)

$A\geqslant 0$ if and only if $PAP^{\top}\geqslant 0$ ; and
(ii)

$A\geqslant 0$ if and only if $D_{v}AD_{v^{-1}}\geqslant 0$ .

The relative gain array was used to give a short proof of the following useful result [17, Lemma 3.3].

Lemma 4.9.

\thlabel

jplemma If $P=P_{\sigma}$ is a permutation matrix and $x\in\mathbb{C}^{n}$ , then $PD_{x}P^{\top}=D_{\sigma(x)}$ .

Theorem 4.10.

\thlabel

conetransforms If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , $P=P_{\sigma}$ is a permutation matrix, and $v$ is a totally nonzero vector, then

(i)

$\mathcal{C}(PS)=\mathcal{C}(S)$ ;
(ii)

$\mathcal{C}(SP)=\sigma^{-1}(\mathcal{C}(S))\coloneqq\{x\in\mathbb{C}^{n}\mid x=\sigma^{-1}(y),~{}y\in\mathcal{C}(S)\}$ ;
(iii)

$\mathcal{C}(D_{v}S)=\mathcal{C}(S)$ , $\forall v>0$ ;
(iv)

$\mathcal{C}(SD_{v})=\mathcal{C}(S)$ ;
(v)

$\mathcal{C}(\alpha S)=\mathcal{C}(S)$ , $\forall\alpha\neq 0$ ;
(vi)

$\mathcal{C}(\bar{S})=\overline{\mathcal{C}(S)}\coloneqq\{y\in\mathbb{C}^{n}\mid y=\bar{x},~{}x\in\mathcal{C}(S)\}$ ;
(vii)

$\mathcal{C}(S^{-1})=\mathcal{C}(S^{\top})$ . In particular, $\mathcal{C}(S)=\mathcal{C}(S^{-\top})$ , where $S^{-\top}\coloneqq(S^{\top})^{-1}=(S^{-1})^{\top}$ ; and
(viii)

$\mathcal{C}(S^{\ast})=\overline{\mathcal{C}(S^{-1})}$ . In particular, $\mathcal{C}(S)=\overline{\mathcal{C}(S^{-\ast})}$ , where $S^{-\ast}\coloneqq(S^{\ast})^{-1}=(S^{-1})^{\ast}$ .

Proof.

(i)

Follows from part (i) of \threfnonnegsims.

(ii)

If $\sigma$ is the permutation corresponding to $P$ , then

	$\displaystyle x\in\mathcal{C}(SP)$	$\displaystyle\Longleftrightarrow(SP)D_{x}(SP)^{-1}\geqslant 0$
		$\displaystyle\Longleftrightarrow S(PD_{x}P^{\top})S^{-1}\geqslant 0$
(\threfjplemma)			$\displaystyle\Longleftrightarrow SD_{y}S^{-1}\geqslant 0,\ y=\sigma(x)$
		$\displaystyle\Longleftrightarrow x=\sigma^{-1}(y),\ y\in\mathcal{C}(S),$
		$\displaystyle\Longleftrightarrow x\in\sigma^{-1}\left(\mathcal{C}(S)\right).$

(iii)

Follows from part (ii) of \threfnonnegsims.

(iv)

Follows from the fact that

SD_{x}S^{-1}=SD_{v\circ x\circ v^{-1}}S^{-1}=S(D_{v}D_{x}D_{v^{-1}})S^{-1}=(SD_{v})D_{x}(SD_{v})^{-1}.

(v)

Immediate from part (iv) since $\alpha S=SD_{\alpha e}$ .
(vi)

Follows from the fact that

$\overline{SD_{x}S^{-1}}=\overline{S}\cdot\overline{D_{x}}\cdot\overline{S^{-1}}=\overline{S}\cdot D_{\bar{x}}\cdot\overline{S}^{-1}.$
(vii)

Follows from the fact that $\left(S^{-1}D_{x}S\right)^{\top}=S^{\top}D_{x}S^{-\top}$ .
(viii)

Immediate from parts (vi) and (vii). ∎

Definition 4.11.

\thlabel

equivrel If $S,T\in\mathsf{GL}_{n}(\mathbb{C})$ , then $S$ is equivalent to $T$ , denoted by $S\sim T$ , if $S=PD_{v}TD_{w}Q$ , where $P=P_{\sigma}$ is a permutation matrix; $Q=Q_{\gamma}$ is a permutation matrix; $v$ is a positive vector; and $w$ is a totally nonzero vector.

Theorem 4.12.

\thlabel

thmequivrel If $\sim$ is defined as in \threfequivrel, then $\sim$ is an equivalence relation on $\mathsf{GL}_{n}(\mathbb{C})$ .

Proof.

If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then it is clear that $S\sim S$ .

If $S=PD_{v}TD_{w}Q$ , then, by \threfjplemma,

\displaystyle T=\left(PD_{v}\right)^{-1}S\left(D_{w}Q\right)^{-1}=D_{v^{-1}}P^{\top}SQ^{\top}D_{w^{-1}}=P_{\sigma^{-1}}D_{\sigma(v^{-1})}SD_{\gamma^{-1}(w^{-1})}Q_{\gamma^{-1}}.

Thus, $T\sim S$ whenever $S\sim T$ .

If $S=PD_{v}TD_{w}Q$ and $T=\hat{P}D_{\hat{v}}UD_{\hat{w}}\hat{Q}$ , then, by \threfjplemma,

\displaystyle S=PD_{v}TD_{w}Q=(PD_{v})(\hat{P}D_{\hat{v}}UD_{\hat{w}}\hat{Q})(D_{w}Q)=(P\hat{P})D_{\hat{\sigma}^{-1}(v)\circ\hat{v}}UD_{\hat{w}\circ\hat{\gamma}(w)}(\hat{Q}Q).

Thus, $S\sim U$ whenever $S\sim T$ and $T\sim U$ . ∎

4.2. Complex Perron Similarities

Definition 4.13.

If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then $S$ is called a Perron similarity if there is an irreducible, nonnegative matrix $A$ and a diagonal matrix $D$ such that $A=SDS^{-1}$ .

Theorem 4.14.

\thlabel

perronsimcharacterization If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then $S$ is a Perron similarity if and only if there is a unique positive integer $k\in\left[n\right]$ such that $Se_{k}=\alpha x$ and $e_{k}^{\top}S^{-1}=\beta y^{\top}$ , where $\alpha$ and $\beta$ are nonzero complex numbers such that $\alpha\beta>0$ , and $x$ and $y$ are positive vectors. Furthermore, if $S$ is a Perron similarity, then $\mathcal{C}(S)$ is nontrivial.

Proof.

If $S$ is a Perron similarity, then there is an irreducible, nonnegative matrix $A$ and a diagonal matrix $D$ such that $A=SDS^{-1}$ . In view of \threfpftirr, there are (possibly empty) diagonal matrices $\hat{D}$ and $\tilde{D}$ such that

D=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\hat{{D}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\\k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\rho(A)$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\tilde{D}$\hfil\kern 5.0pt\crcr}}}}\right]$}},\ 1\leqslant k\leqslant n,

where $\rho(A)\notin\sigma(\hat{D})$ and $\rho(A)\notin\sigma(\tilde{D})$ . If $s_{k}\coloneqq Se_{k}$ , then $As_{k}=\rho(A)s_{k}$ since $AS=SD$ . Because the geometric multiplicity of an eigenvalue is less than or equal to its algebraic multiplicity [13, p. 181], it follows that $\dim\mathsf{E}_{\rho(A)}=1$ . Hence, there is a nonzero complex number $\alpha$ such that $s_{k}=\alpha x$ , where $x$ denotes the unique right Perron vector of $A$ .

Because the line of reasoning above applies to $A^{\top}=(S^{-\top})DS^{\top}$ , it follows that there is a nonzero complex number $\beta$ such that $t_{k}=\beta y$ , where $t_{k}^{\top}\coloneqq e_{k}^{\top}S^{-1}$ and $y$ denotes the unique left Perron vector of $A$ .

Since $S^{-1}S=I$ , it follows that $1=\left(e_{k}^{\top}S^{-1}\right)\left(Se_{k}\right)=t_{k}^{\top}s_{k}=(\alpha\beta)y^{\top}x$ . As $x$ and $y$ are positive, we obtain $\alpha\beta=(y^{\top}x)^{-1}>0$ .

Conversely, if there is a positive integer $k$ such that $Se_{k}=\alpha x$ and $e_{k}^{\top}S^{-1}=\beta y^{\top}$ , where $\alpha$ and $\beta$ are nonzero complex numbers such that $\alpha\beta>0$ , and $x$ and $y$ are positive vectors, then $M_{e_{k}}=SD_{e_{k}}S^{-1}=S(e_{k}e_{k}^{\top})S^{-1}=(Se_{k})(e_{k}^{\top}S^{-1})=(\alpha x)(\beta y^{\top})=(\alpha\beta)xy^{\top}>0$ . Thus, $S$ is a Perron similarity.

For uniqueness, suppose, for contradiction, that there is a positive integer $\ell\neq k$ such that $s_{\ell}\coloneqq Se_{\ell}=\gamma u$ and $t_{\ell}^{\top}\coloneqq e_{\ell}^{\top}S^{-1}=\delta v^{\top}$ , where $\gamma$ and $\delta$ are nonzero complex numbers such that $\gamma\delta>0$ , and $u$ and $v$ are positive vectors. Since $AS=SD$ , it follows that $s_{\ell}$ and $t_{\ell}$ are right and left eigenvectors corresponding to an eigenvalue $\lambda\neq\rho(A)$ . By the principle of biorthogonality [13, Theorem 1.4.7(a)], $t_{\ell}^{\ast}s_{k}=0$ . However,

t_{\ell}^{\ast}s_{k}=\overline{t_{\ell}^{\top}}s_{k}=\overline{\delta v^{\top}}(\alpha x)=(\bar{\delta}\alpha)v^{\top}x\neq 0,

a contradiction.

Finally, suppose that $S$ is a Perron similarity. For contradiction, if $\mathcal{C}(S)$ is trivial, then $\mathcal{A}(S)$ is trivial and only contains nonnegative matrices of the form $\alpha I$ , a contradiction if $S$ is a Perron similarity. ∎

Remark 4.15.

It is known that if $\mathcal{C}(S)$ is nontrivial, then $S$ is not necessarily a Perron similarity (see Dockter et al. [7, Example 11]).

4.3. Normalization

If $A$ is a real matrix, then any eigenvector associated with a real eigenvalue of $A$ may be taken to be real ([13, p. 48, Problem 1.1.P3]), and if $v$ is an eigenvector corresponding to a nonreal eigenvalue $\lambda$ of $A$ , then $\bar{v}$ is an eigenvector corresponding to $\bar{\lambda}$ ([13, p. 45]). In view of these elementary facts, and taking into account parts (i)–(iv) of \threfconetransforms and \threfthmequivrel, if $S$ is a Perron similarity, then

(4.5)

S\sim\begin{bmatrix}e&s_{2}&\cdots&s_{r}&s_{r+1}&\bar{s}_{r+1}&\cdots&s_{c}&\bar{s}_{c}\end{bmatrix},

where $s_{i}\in S^{n}$ , $i=2,\dots,c$ ; $\imaginary(s_{i})=0$ , for $i=2,\dots,r$ ; and $\imaginary(s_{i})\neq 0$ , for $i=r+1,\dots,c$ . Hereinafter it is assumed that every Perron similarity is normalized.

The following simple, but useful fact was shown for real matrices [19, Lemma 2.1]. Although the proof extends to complex matrices without alteration, a demonstration is included for completeness.

Proposition 4.16.

\thlabel

simple If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then $x^{\top}S^{-1}\geqslant 0$ if and only if $x\in\mathcal{C}_{r}(S)$ .

Proof.

Notice that

y^{\top}\coloneqq x^{\top}S^{-1}\geqslant 0\Longleftrightarrow x^{\top}=y^{\top}S,~{}y\geqslant 0\Longleftrightarrow x\in\mathcal{C}_{r}(S).\qed

Definition 4.17.

Given an $m$ -by- $n$ matrix $S$ , the row cone of $S$ [19] denoted by $\mathcal{C}_{r}(S)$ , is the the polyhedral cone generated by the rows of $S$ , i.e., $\mathcal{C}_{r}(S)\coloneqq\operatorname{coni}(r_{1},\dots,r_{m})$ and the row polytope of $S$ , denoted by $\mathcal{P}_{r}(S)$ , is the polytope generated by the rows of $S$ , i.e., $\mathcal{P}_{r}(S)\coloneqq\operatorname{conv}(r_{1},\dots,r_{m})$ .

Johnson and Paparella [17] demonstrated that $\mathcal{C}(S)$ can coincide with $\mathcal{C}_{r}(S)$ for a class of Hadamard matrices called Walsh matrices (see \threfwalshmatrices below). We extend and generalize these results for complex matrices.

Definition 4.18.

If $S$ is a Perron similarity, then $S$ is called ideal if $\mathcal{C}(S)=\mathcal{C}_{r}(S)$ .

Lemma 4.19.

\thlabel

realizablerows If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then $\mathcal{C}_{r}(S)\subseteq\mathcal{C}(S)$ if and only if $r_{i}\in\mathcal{C}(S),\ \forall i\in\left[n\right]$ .

Proof.

Immediate from \threfconecontain. ∎

Lemma 4.20.

\thlabel

allonesrow If $S\in\mathsf{GL}_{n}(\mathbb{C})$ , then $\mathcal{C}(S)\subseteq\mathcal{C}_{r}(S)$ if and only if $e\in\mathcal{C}_{r}(S)$ .

Proof.

The necessity of the condition is trivial given that $e\in\mathcal{C}(S)$ .

For sufficiency, assume that $e\in\mathcal{C}_{r}(S)$ and let $x\in\mathcal{C}(S)$ . By definition, there is a nonnegative vector $y$ such that $e^{\top}=y^{\top}S$ and $SD_{x}S^{-1}\geqslant 0$ . Since

x^{\top}S^{-1}=(e^{\top}D_{x})S^{-1}=((y^{\top}S)D_{x})S^{-1}=y^{\top}(SD_{x}S^{-1})\geqslant 0,

it follows that $x\in\mathcal{C}_{r}(S)$ by \threfsimple . ∎

Theorem 4.21.

\thlabel

thm:idealsims If $S$ is a Perron similarity, then $S$ is ideal if and only if $e\in\mathcal{C}_{r}(S)$ and $r_{i}\in\mathcal{C}(S),\ \forall i\in\left[n\right]$ .

Proof.

Immediate from Lemmas LABEL:realizablerows and LABEL:allonesrow. ∎

Theorem 4.22.

\thlabel

coneandpoly If $S$ is a Perron similarity, then $S$ is ideal if and only if $\mathcal{P}_{r}(S)=\mathcal{P}(S)$ .

Proof.

If $S$ is ideal, then $\mathcal{C}_{r}(S)=\mathcal{C}(S)$ . If $x\in\mathcal{P}_{r}(S)$ , then $x\in\mathcal{C}_{r}(S)$ and, by hypothesis, $x\in\mathcal{C}(S)$ . Thus, $M_{x}\geqslant 0$ and it suffices to demonstrate that $M_{x}e=e$ . Since $Se_{1}=e$ , it follows that $S^{-1}e=e_{1}$ . Furthermore, any convex combination of the rows of $S$ produces a vector whose first entry is 1. Thus, $x_{1}=1$ and

\displaystyle M_{x}e=(SD_{x}S^{-1})e=SD_{x}e_{1}=Se_{1}=e,

i.e., $x\in\mathcal{P}(S)$ . If $x\in\mathcal{P}(S)$ , then $M_{x}\geqslant 0$ and $M_{x}e=e$ . Since $Se_{1}=e$ and $M_{x}$ has a positive eigenvector, it follows that $x_{1}=\rho(M_{x})=1$ [13, Corollary 8.1.30]. Notice that $x\in\mathcal{P}(S)\implies x\in\mathcal{C}(S)\implies x\in\mathcal{C}_{r}(S)$ , i.e., $\exists y\geqslant 0$ such that $x^{\top}=y^{\top}S$ . Thus,

y^{\top}e=(x^{\top}S^{-1})e=x^{\top}(S^{-1}e)=x^{\top}e_{1}=x_{1}=1,

i.e., $x\in\mathcal{P}_{r}(S)$ .

Conversely, suppose that $\mathcal{P}_{r}(S)=\mathcal{P}(S)$ . Since $e\in\mathcal{C}(S)$ , it follows from Propositions LABEL:exptball and LABEL:exptconv that one of the rows of $S$ must be $e^{\top}$ . Thus, $\mathcal{C}(S)\subseteq\mathcal{C}_{r}(S)$ by \threfallonesrow. By hypothesis, every row of $S$ is realizable so that, following Lemma LABEL:realizablerows, $\mathcal{C}_{r}(S)\subseteq\mathcal{C}(S)$ . ∎

Corollary 4.23.

If $S$ is a Perron similarity, then $S$ is ideal if and only if $r_{i}\in\mathcal{C}(S)$ , $\forall i\in\left[n\right]$ , and $\exists k\in\left[n\right]$ such that $e_{k}^{\top}S=e^{\top}$ .

Proof.

Immediate from \threfcor:allonesextreme and \threfthm:idealsims. ∎

Definition 4.24.

If $S$ is a Perron similarity, then $S$ is called extremal if $\mathcal{P}(S)$ contains an extremal point other than $e$ .

4.4. Kronecker Product & Walsh Matrices

If $A\in\mathsf{M}_{m\times n}(\mathbb{F})$ and $B\in\mathsf{M}_{p\times q}(\mathbb{F})$ , then the Kronecker product of $A$ and $B$ , denoted by $A\otimes B$ , is the $mp$ -by- $nq$ matrix defined blockwise by $A\otimes B=\begin{bmatrix}a_{ij}B\end{bmatrix}$ .

If $A\in\mathsf{M}_{m\times n}(\mathbb{F})$ and $p\in\mathbb{N}_{0}$ , then

A^{\otimes p}\coloneqq\begin{cases}[1],&p=0\\ A^{\otimes(p-1)}\otimes A=A\otimes A^{\otimes(p-1)},&p>1.\end{cases}

Although some of the definitions differ, the proofs for the following results, established by Dockter et al. [7], can be retooled to obtain the following.

Theorem 4.25 ([7, Theorem 7]).

\thlabel

kronPS If $S$ and $T$ are Perron similarities, then $S\otimes T$ is a Perron similarity.

Theorem 4.26 ([7, Theorem 13]).

\thlabel

kronideal If $S$ and $T$ are ideal, then $S\otimes T$ is ideal.

Definition 4.27.

If $H=[h_{ij}]\in\mathsf{M}_{n}(\{\pm 1\})$ , then $H$ is called a Hadamard matrix if $HH^{\top}=nI_{n}$ .

Definition 4.28.

\thlabel

walshmatrices If $n\in\mathbb{N}_{0}$ , then the matrix

H_{2^{n}}\coloneqq\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}^{\otimes n}

is called Sylvester’s Hadamard matrix or, for brevity, the Walsh matrix (of order $2^{n}$ ).

It is well-known that $H_{2^{n}}$ is a Hadamard matrix. Notice that $H_{1}=[1]$ , $H_{2}=\left[\begin{array}[]{rr}1&1\\ 1&-1\end{array}\right]$ and

H_{4}=\left[\begin{array}[]{*{4}{r}}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\end{array}\right].

The theory of association schemes was used to prove that these matrices are ideal [17, Proposition 5.1 and Theorem 5.2]. However, a proof of this is readily available via induction coupled with Theorems LABEL:kronPS and LABEL:kronideal.

Theorem 4.29.

If $n\in\mathbb{N}_{0}$ , then $H_{2^{n}}$ is ideal, extremal, and $\dim(\mathcal{C}(H_{2^{n}}))=2^{n}$ .

Although every normalized Hadamard matrix is a Perron similarity, not every Hadamard matrix is ideal (it can be verified via the MATLAB-command hadamard(12) that only the first row, i.e., the all-ones vector, of the normalized Hadamard matrix of order twelve is realizable). However, it is known that if $H$ is a Hadamard matrix, then $\operatorname{conv}(e,e_{1}-e_{2},\ldots,e_{1}-e_{n})\subseteq\mathcal{P}(H)$ [17, Remark 6.4]. Furthermore, in view of \threfallonesrow, we have

\operatorname{conv}(e,e_{1}-e_{2},\ldots,e_{1}-e_{n})\subseteq\mathcal{P}(H)\subseteq\mathcal{P}_{r}(H).

Thus, every Hadamard matrix is extremal.

If $x\in\mathbb{C}^{n}$ and $\sigma_{1},\ldots,\sigma_{n}\in\mathsf{Sym}(n)$ , then a matrix of the form

X=\begin{bmatrix}\sigma_{1}(x)^{\top}\\ \vdots\\ \sigma_{n}(x)^{\top}\end{bmatrix}

is called a permutative matrix. Notice that permutation matrices and circulant matrices are permutative matrices.

If $y=Hx$ and $M_{y}=2^{-n}H_{2^{n}}D_{y}H_{2^{n}}$ , then $A$ is a permutative matrix [17, p. 295]. For example, if $n=2$ , then

M_{y}=\begin{bmatrix}x_{1}&x_{2}\\ x_{2}&x_{1}\end{bmatrix}

and when $n=4$ ,

(4.6)

M_{y}=\begin{bmatrix}x_{1}&x_{2}&x_{3}&x_{4}\\ x_{2}&x_{1}&x_{4}&x_{3}\\ x_{3}&x_{4}&x_{1}&x_{2}\\ x_{4}&x_{3}&x_{2}&x_{1}\end{bmatrix}.

In general, let

P_{1}^{(1)}:=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\mbox{ and }P_{2}^{(1)}:=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}.

and for $n\geq 2$ , let

P_{n,k}:=\left\{\begin{array}[]{rl}\begin{bmatrix}P_{(n-1),k}&0\\ 0&P_{(n-1),k}\end{bmatrix}\in\mathsf{M}_{2^{n}}(\mathbb{R}),&k\in\left[2^{n-1}\right]\\ \\ \begin{bmatrix}0&P_{(n-1),k-2^{n-1}}\\ P_{(n-1),k-2^{n-1}}&0\end{bmatrix}\in\mathsf{M}_{2^{n}}(\mathbb{R}),&k\in\left[2^{n}\right]\backslash\left[2^{n-1}\right],\end{array}\right.

If $y=Hx$ , then

(4.7)

M_{y}=\begin{bmatrix}x^{\top}P_{n,1}\\ \vdots\\ x^{\top}P_{n,2^{k}}\end{bmatrix}.

Kalman and White [20] called the matrix (4.6) a Klein matrix. As such, we call any matrix of the form given in (4.7) a Klein matrix of order $2^{n}$ .

5. Circulant Matrices

In what follows, we let $F=F_{n}$ denote the $n$ -by- $n$ discrete Fourier transform matrix, i.e., $F$ is the $n$ -by- $n$ matrix such that

f_{ij}=\frac{1}{\sqrt{n}}\omega^{(i-1)(j-1)},

with

\omega=\omega_{n}\coloneqq\cos\left(\frac{2\pi}{n}\right)-\sin\left(\frac{2\pi}{n}\right)\mathsf{i}.

If $n>1$ , then

\displaystyle F

\displaystyle=\frac{1}{\sqrt{n}}\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n\\1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{k-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{n-1}\\\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\ddots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots\\k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{k-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(k-1)^{2}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(k-1)(n-1)}\\\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\ddots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots\\n$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{n-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(k-1)(n-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(n-1)(n-1)}$\hfil\kern 5.0pt\crcr}}}}\right]$}}\\

\displaystyle=\frac{1}{\sqrt{n}}\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{\omega}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{\omega}^{k-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{\omega}^{n-1}$\hfil\kern 5.0pt\crcr}}}}\right]$}},

with

v_{\omega}\coloneqq\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\\1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega\\\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots\\k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{k-1}\\\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots\\n$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{n-1}$\hfil\kern 5.0pt\crcr}}}}\right]$}}.

Because $F$ is symmetric and unitary [5, Theorem 2.5.1], it follows that $F^{-1}=F^{\ast}=\overline{F^{\top}}=\overline{F}$ .

Since $\omega^{k}\cdot\omega^{n-k}=1=\omega^{k}\cdot\overline{\omega^{k}}$ , it follows that $\omega^{n-k}=\overline{\omega^{k}}$ . Consequently,

(5.1)

v_{\omega}^{n-k}=\overline{v_{w}^{k}},\ 1\leqslant k\leqslant n-1.

Thus,

F=\frac{1}{\sqrt{n}}\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 3$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{\omega}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{\omega}^{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\overline{v_{w}^{2}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\overline{v_{w}}$\hfil\kern 5.0pt\crcr}}}}\right]$}}.

Definition 5.1 ([5, p. 66]).

If $c=\begin{bmatrix}c_{1}&\cdots&c_{n}\end{bmatrix}^{\top}\in\mathbb{C}^{n}$ and $C\in\mathsf{M}_{n}(\mathbb{C})$ is a matrix such that

c_{ij}=c_{((j-i)\bmod{n})+1},\ \forall(i,j)\in\left[n\right]^{2},

then $C$ is called a circulant or a circulant matrix with reference vector $c$ . In such a case, we write $C=\operatorname{circ}(c)=\operatorname{circ}(c_{1},\ldots,c_{n})$ .

If $C\in\mathsf{M}_{n}(\mathbb{C})$ , then $C$ is circulant if and only if there is a diagonal matrix $D$ such that $C=FDF^{\ast}=FD\overline{F}$ [5, Theorems 3.2.2 and 3.2.3].

Recall that $A=[A_{ij}]$ , with $A_{ij}\in\mathsf{M}_{n}(\mathbb{C})$ is called (an $m$ -by- $m$ ) block matrix. If $C=\operatorname{circ}(C_{1},\ldots,C_{m})$ with $C_{k}\in\mathsf{M}_{n}(\mathbb{C}),\ \forall k\in\left[m\right]$ , then the block matrix $C$ is called block-circulant or a block-circulant matrix of type $(m,n)$ [5, §5.6]. The set of block-circulant matrices of type $(m,n)$ is denoted by $\mathcal{BC}_{mn}$ .

If $C=[C_{ij}]$ is an $m$ -by- $m$ block matrix and $C_{ij}$ is circulant, then $C$ is called a circulant block matrix and the set of such matrices is denoted by $\mathcal{CB}_{mn}$ [5, §5.7].

Combining the previous sets yields the class of block circulant matrices with circulant blocks, i.e., block matrices of the form $C=\operatorname{circ}(C_{1},\ldots,C_{m})$ , where $C_{1},\ldots,C_{m}$ are circulant. The set of such matrices is denoted by $\mathcal{BCCB}_{mn}$ . All matrices in $\mathcal{BCCB}_{mn}$ are simultaneously diagonalizable by the unitary matrix $F_{m}\otimes F_{n}$ and any matrix of the form

(F_{m}\otimes F_{n})D(\overline{F_{m}}\otimes\overline{F_{n}}),

with $D$ diagonal, belongs to $\mathcal{BCCB}_{mn}$ [5, Theorem 5.8.1].

6. Perron Similarities arising from K-arcs

In this section, we examine Perron similarities from realizing matrices of points on Type I K-arcs. Attention is focused on Type I K-arcs because many Type II arcs and Type III arcs are pointwise powers of Type I arcs (see, e.g., Munger et al. [27, Section 5]).

6.1. Type 0

Points on the Type 0 arc are zeros of the reduced Ito polynomial $f_{\alpha}(t)=(t-\beta)^{n}-\alpha^{n}$ , where $\alpha\in[0,1]$ and $\beta\coloneqq 1-\alpha$ . In [18] it was showed (and it is easy to verify otherwise) that the matrix

M=M(\alpha)\coloneqq\begin{bmatrix}\beta&\alpha&\\ &\beta&\alpha\\ &&\ddots&\ddots\\ &&&\beta&\alpha\\ \alpha&&&&\beta\end{bmatrix}

realizes this arc, i.e., $\chi_{M}=f_{\alpha}$ . Because $M$ is a circulant, there is a diagonal matrix $D$ such that $M=FDF^{\ast}$ . As $F$ is a scaled Vandermonde matrix, we defer its discussion to the subsequent section.

6.2. Type I

For Type I arcs, the reduced Ito polynomial is $f_{\alpha}(t)=t^{s}-\beta t^{s-q}-\alpha$ , with $\alpha\in[0,1]$ and $\beta\coloneqq 1-\alpha$ . If

M=M(\alpha)\coloneqq\begin{bmatrix}0&I\\ \alpha&\beta e_{s-q}^{\top}\end{bmatrix},

then $M$ is a nonnegative irreducible companion matrix and $\chi_{M}=f_{\alpha}$ . Following Remark 3.14, there are at most $2s-1$ complex values, and hence at most $2s-1$ values in $[0,1]$ , such that $f_{\alpha}$ does not have distinct roots. Notice that $f_{\alpha}(1)=0$ and if $1,\lambda_{2},\dots,\lambda_{s}$ are the distinct zeros of $f_{\alpha}$ , then the Vandermonde matrix

(6.1)

S=S(\alpha)=\begin{bmatrix}1&1&\cdots&1\\ 1&\lambda_{2}&\cdots&\lambda_{s}\\ \vdots&\vdots&\ddots&\vdots\\ 1&\lambda_{2}^{s-1}&\cdots&\lambda_{s}^{s-1}\end{bmatrix}\in\mathsf{GL}_{s}(\mathbb{C})

is a Perron similarity satisfying $M=S\operatorname{diag}{\left(1,\lambda_{2},\dots,\lambda_{s}\right)}S^{-1}$ . The Perron similarity $S$ is ideal because every row is realizable (the $k\textsuperscript{th}$ -row forms the spectrum of $M^{k}\geqslant 0$ ) and the first row is $e$ . Furthermore, it is extremal because the second row is extremal.

Corollary 6.1.

\thlabel

TypeIsims Let $f_{\alpha}$ be defined as in (3.10), and let $S=S(\alpha)$ be the Vandermonde matrix defined as in $\eqref{vandermonde}$ corresponding to the zeros of $f_{\alpha}$ .

(i)

If $n$ is even, then $S$ is ideal and extremal.
(ii)

If $n$ is odd and $\alpha\geqslant\beta$ , then $S$ is ideal and extremal.
(iii)

If $n$ is odd, $\alpha<\beta$ , and $(n-1)^{n-1}\beta^{n}-n^{n}\alpha^{n-1}\neq 0$ , then $S$ is ideal and extremal.

7. Circulant and Block-Circulant NIEP

Since $S\sim\alpha S$ , with a slight abuse of notation we let $F=F_{n}=[f_{ij}]$ denote the $n$ -by- $n$ matrix such that $f_{ij}=\omega^{(i-1)(j-1)}$ . Notice that $F$ is a Vandermonde matrix corresponding to the $n$ distinct $n\textsuperscript{th}$ -roots of unity, i.e,

\displaystyle F

\displaystyle=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n\\1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{k-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{n-1}\\\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\ddots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots\\k$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{k-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(k-1)^{2}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(k-1)(n-1)}\\\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\ddots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\vdots\\n$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{n-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(k-1)(n-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\omega^{(n-1)^{2}}$\hfil\kern 5.0pt\crcr}}}}\right]$}}\\

\displaystyle=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 3$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{\omega}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{\omega}^{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\overline{v_{w}^{2}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\overline{v_{w}}$\hfil\kern 5.0pt\crcr}}}}\right]$}},\ n>1.

It is easy to show that

(7.1)

F^{-1}=\frac{1}{n}\overline{F},

(7.2)

(F_{m}\otimes F_{n})^{-1}=\frac{1}{mn}\left(\overline{F_{m}}\otimes\overline{F_{n}}\right),

and $C\in\mathsf{M}_{n}(\mathbb{C})$ is circulant if and only if there is a diagonal matrix $D$ such that $C=FDF^{-1}$ .

Furthermore, if $y\in\mathbb{C}^{n}$ , $x=Fy$ , and $M_{x}=FD_{x}F^{-1}$ , then, following (4.4),

(7.3)

[M_{x}]_{1,j}=\left\langle x,\overline{e\circ\frac{\overline{f_{j}}}{n}}\right\rangle=\left\langle x,\frac{f_{j}}{n}\right\rangle=\frac{1}{n}\overline{f_{j}}^{\top}(Fy)=y_{j},\ \forall j\in\left[n\right]

i.e., $M_{x}=\operatorname{circ}y$ .

Corollary 7.1 (extreme ray and vertex representation).

\thlabel

dftconetope If $n\in\mathbb{N}$ , then $F$ is ideal, extremal, and $\dim\mathcal{C}(F)=n$ .

Proof.

The claim that $F$ is ideal is immediate from \threfTypeIsims or (7.3). Clearly, $F$ is extremal since every entry of $F$ is extremal in $\Theta_{n}$ . Finally, $\dim\mathcal{C}(F)=n$ because $F$ is invertible. ∎

Theorem 7.2 (half-space description).

If $n$ is a positive integer, then

\mathcal{C}(F)=\bigcap_{k\in\left[n\right]}\mathsf{H}_{k},

where

\mathsf{H}_{k}\coloneqq\mathsf{H}({f_{k}})\cap\mathsf{H}(\mathsf{i}{f_{k}})\cap\mathsf{H}(-\mathsf{i}{f_{k}})

and $f_{k}\coloneqq Fe_{k}$ , $k\in\left[n\right]$ .

Proof.

If $x\in\mathcal{C}(F)$ , then, by \threfdftconetope, there is a nonnegative vector $y$ such that $x^{\top}=y^{\top}F$ . Since $F$ is symmetric, it follows that $x=Fy$ . Since

(7.4)

\langle x,f_{k}\rangle=\left(Fe_{k}\right)^{\ast}(Fy)=\left(e_{k}^{\top}\overline{F}\right)(Fy)=e_{k}^{\top}\left(\left(\overline{F}F\right)y\right)=e_{k}^{\top}((nI_{n})y)=ny_{k},

it follows that $\real(\langle x,{f_{k}}\rangle)\geqslant 0$ and $x\in\mathsf{H}(f_{k})$ . Because $\langle x,\pm\mathsf{i}f_{k}\rangle=\mp\mathsf{i}\langle x,f_{k}\rangle=\mp\mathsf{i}ny_{k}$ , it follows that $\real(\langle x,\pm\mathsf{i}f_{k}\rangle)\geqslant 0$ , i.e., $x\in\mathsf{H}(\pm\mathsf{i}f_{k})$ . As $k$ and $x$ were arbitrary, $\mathcal{C}(F)\subseteq\bigcap_{k\in\left[n\right]}\mathsf{H}_{k}$ .

Let $x\in\bigcap_{k\in\left[n\right]}\mathsf{H}_{k}$ and suppose, for contradiction, that $x\notin\mathcal{C}(F)$ . Because $F$ is invertible, there is a unique complex vector $y$ such that $x=Fy$ , and since $F$ is symmetric, we have $x^{\top}=y^{\top}F$ . As $x\notin\mathcal{C}(F)$ , it follows that $a\coloneqq\real y\not\geqslant 0$ or $b\coloneqq\imaginary y\neq 0$ . Thus, $\exists k\in\left[n\right]$ such that $\real y_{k}=a_{k}<0$ or $\imaginary y_{k}=b_{k}\neq 0$ . Notice that $\langle x,f_{k}\rangle=ny_{k},\ \forall k\in\left[n\right]$ because the calculations in (7.4) do not rely on the nonnegativity of $y$ . If $a_{k}<0$ , then $\langle x,f_{k}\rangle=ny_{k}=n(a_{k}+\mathsf{i}b_{k})$ implies $\real(\langle x,f_{k}\rangle)=na_{k}<0$ , a contradiction since $x\in\mathsf{H}(f_{k})$ . Otherwise, if $b_{k}\neq 0$ , then $b_{k}<0$ or $b_{k}>0$ ; if $b_{k}<0$ , then the equation

\langle x,\mathsf{i}f_{k}\rangle=-\mathsf{i}\langle x,f_{k}\rangle=-\mathsf{i}ny=n(b_{k}-\mathsf{i}a_{k})

implies $\real(\langle x,\mathsf{i}f_{k}\rangle)=nb_{k}<0$ , a contradiction since $x\in\mathsf{H}(\mathsf{i}f_{k})$ ; if $b_{k}>0$ , then the equation

\langle x,-\mathsf{i}f_{k}\rangle=\mathsf{i}\langle x,f_{k}\rangle=\mathsf{i}ny=n(-b_{k}+\mathsf{i}a_{k})

implies $\real(\langle x,-\mathsf{i}f_{k}\rangle)=-nb_{k}<0$ , a contradiction since $x\in\mathsf{H}(-\mathsf{i}f_{k})$ . Thus, $\mathcal{C}(F)\supseteq\bigcap_{k\in\left[n\right]}\mathsf{H}_{k}$ and the demonstration is complete. ∎

Proposition 7.3.

\thlabel

propF If $n\in\mathbb{N}$ , then:

(i)

$\mathcal{C}(F)=\mathcal{C}\left(\overline{F}\right)$ ;
(ii)

$\overline{F}$ is ideal; and
(iii)

$\mathcal{P}(\overline{F})=\mathcal{P}_{r}(\overline{F})$ .

Proof.

(i)

Notice that

\displaystyle\mathcal{C}\left(\overline{F}\right)=\mathcal{C}\left(\frac{1}{n}\overline{F}\right)=\mathcal{C}(F^{-1})=\mathcal{C}(F^{\top})=\mathcal{C}(F).

(ii)

Follows from the observation that

	$\displaystyle y\in\mathcal{C}\left(\overline{F}\right)$	$\displaystyle\iff y\in\overline{\mathcal{C}(F)}$
		$\displaystyle\iff y=\overline{x},\ x\in\mathcal{C}(F)$
		$\displaystyle\iff y=\overline{x},\ x=Fz,\ z\geqslant 0$
		$\displaystyle\iff y=\overline{F}z,\ z\geqslant 0$
		$\displaystyle\iff y\in\mathcal{C}_{r}(\overline{F}).$

(iii)

The assertion that $\mathcal{P}(\overline{F})=\mathcal{P}_{r}(\overline{F})$ is an immediate consequence of part (ii) and \threfconeandpoly. ∎

There is an easier test for feasibility, as follows.

Theorem 7.4.

If $x\in\mathbb{C}^{n}$ , then $\Lambda(x)$ is realizable by an $n$ -by- $n$ circulant matrix if and only if $Fx\geqslant 0$ .

Proof.

If $\Lambda(x)$ is realizable by an $n$ -by- $n$ circulant matrix $C$ , then $x\in\mathcal{C}(F)$ . By parts (i) and (ii) of \threfpropF, there is a nonnegative vector $y$ such that $x^{\top}=y^{\top}\overline{F}$ . Note that $x=\overline{F}y$ since $\overline{F}$ is symmetric and because $F\overline{F}=nI_{n}$ , we have $Fx=ny\geqslant 0$ .

Conversely, if $y\coloneqq Fx\geqslant 0$ , then

x=\frac{1}{n}\overline{F}y

and since $\overline{F}$ is symmetric, it follows that

x^{\top}=\frac{1}{n}y^{\top}\overline{F}.

By parts (i) and (ii) of \threfpropF, $x\in\mathcal{C}({G})$ , i.e., $\Lambda(x)$ is realizable by an $n$ -by- $n$ circulant matrix. ∎

Corollary 7.5.

If $x\in\mathbb{C}^{n}$ , then $\Lambda(x)$ is realizable by an $n$ -by- $n$ circulant matrix if and only if

(7.5)

\sum_{j=1}^{n}\omega^{(i-1)(j-1)}x_{j}\geqslant 0,\ \forall i\in\left[n\right].

If $x\in\mathbb{C}^{n}$ satisfies the inequalities in (7.5), then the inequality corresponding to $i=1$ in yields

\sum_{k=1}^{n}x_{k}\geqslant 0

corresponding to the trace-condition.

Remark 7.6.

\thlabel

rem_symm For $n\geqslant 2$ , let

\mathbb{C}_{\rm sym}^{n}\coloneqq\{x\in\mathbb{C}^{n}\mid\imaginary x_{1}=0,\ \overline{x_{k}}=x_{n-k+2},\ 2\leqslant k\leqslant n\}.

In the light of (5.1), notice that

g_{n-k+2}=v_{\omega}^{n-k+1}=\overline{v_{\omega}^{k-1}}=\overline{f_{k}},\ 2\leqslant k\leqslant n.

As such, the vector $Gx$ is real if and only if $x\in\mathbb{C}_{\rm sym}^{n}$ .

Example 7.7.

In view of the inequalities given by (7.5) and \threfrem_symm, if $x\in\mathbb{C}^{2}$ and $\Lambda(x)=\{x_{1},x_{2}\}$ , then $\Lambda$ is realizable by a circulant matrix if and only if $x_{1},x_{2}\in\mathbb{R}$ and

\displaystyle\left\{\begin{array}[]{l}x_{1}+x_{2}\geqslant 0\\ x_{1}-x_{2}\geqslant 0\end{array}\right..

If $x\in\mathbb{C}^{4}$ and $\Lambda(x)=\{x_{1},x_{2},x_{3},x_{4}\}$ , then $\Lambda$ is realizable by a circulant matrix if and only if $x_{1},x_{3}\in\mathbb{R}$ , $x_{4}=\overline{x_{2}}$ , and

\displaystyle\left\{\begin{array}[]{*{9}{l}}x_{1}&+&x_{2}&+&x_{3}&+&\overline{x_{2}}&\geqslant&0\\ x_{1}&-&x_{2}i&-&x_{3}&+&\overline{x_{2}}i&\geqslant&0\\ x_{1}&-&x_{2}&+&x_{3}&-&\overline{x_{2}}&\geqslant&0\\ x_{1}&+&x_{2}i&-&x_{3}&-&\overline{x_{2}}i&\geqslant&0\end{array}\right..

Theorem 7.8.

If $m,n\in\mathbb{N}$ , then $F_{m}\otimes F_{n}$ is ideal, extremal, and $\dim\mathcal{C}(F_{m}\otimes F_{n})=mn$ .

Proof.

Immediate from \threfkronideal and \threfdftconetope. ∎

Theorem 7.9 (half-space description).

If $m,n\in\mathbb{N}$ , then

\mathcal{C}(F_{m}\otimes F_{n})=\bigcap_{k\in\left[n\right]}\mathsf{H}_{k},

where

\mathsf{H}_{k}\coloneqq\mathsf{H}({f_{k}})\cap\mathsf{H}(\mathsf{i}{f_{k}})\cap\mathsf{H}(-\mathsf{i}{f_{k}})

and $f_{k}\coloneqq(F_{m}\otimes F_{n})e_{k}$ , $k\in\left[mn\right]$ .

Proposition 7.10.

If $m,n\in\mathbb{N}$ , then:

(i)

$\mathcal{C}({F_{m}}\otimes{F_{n}})=\mathcal{C}\left(\overline{F_{m}}\otimes\overline{F_{n}}\right)$ ;
(ii)

$\overline{F_{m}}\otimes\overline{F_{n}}$ is ideal; and
(iii)

$\mathcal{P}(\overline{F_{m}}\otimes\overline{F_{n}})=\mathcal{P}_{r}(\overline{F_{m}}\otimes\overline{F_{n}})$ .

Proof.

Analogous to the proof of \threfpropF using (7.2) and properties of the Kronecker product. ∎

Corollary 7.11.

If $x\in\mathbb{C}^{n}$ , then $\Lambda(x)$ is realizable by an $(m,n)$ block-circulant matrix with circulant blocks if and only if $(F_{m}\otimes F_{n})x\geqslant 0$ .

Theorem 7.12.

If $k\in\mathbb{N}_{0}$ and $n\in\mathbb{N}$ , then $F_{n}\otimes H_{2^{k}}$ is ideal, extremal, and $\dim\mathcal{C}(F_{n}\otimes H_{2^{k}})=2^{k}n$

Corollary 7.13.

If $x\in\mathbb{C}^{n}$ , then $\Lambda(x)$ is realizable by a Klein block matrix with circulant blocks if and only if $(F_{n}\otimes H_{2^{k}})x\geqslant 0$ .

Remark 7.14.

If $A\in\mathsf{M}_{m}(\mathbb{C})$ and $B\in\mathsf{M}_{n}(\mathbb{C})$ , then there is a permutation matrix $P$ such that $A\otimes B=P(B\otimes A)P^{\top}$ [12, Corollary 4.3.10]. As a consequence, $F_{m}\otimes F_{n}\sim F_{n}\otimes F_{m}$ . Furthermore, if $m$ and $n$ are relatively prime, then there are permutation matrices $P$ and $Q$ such that $F_{mn}=P(F_{m}\otimes F_{n})Q$ [10, 29]. Thus, $F_{mn}\sim F_{m}\otimes F_{n}$ whenever $\gcd(m,n)=1$ . It is also clear that $F_{mn}\not\sim F_{m}\otimes F_{n}$ whenever $\gcd(m,n)>1$ .

Recall that if $A_{1},\ldots,A_{m}$ are matrices with $A_{k}\in\mathsf{M}_{m_{k}\times n_{k}}(\mathbb{C})$ , then

\bigotimes_{k=1}^{m}A_{k}\coloneqq\begin{cases}A_{1},&k=1\\ \left(\bigotimes_{k=1}^{m-1}A_{k}\right)\otimes A_{m},&k>1.\end{cases}

Theorem 7.15.

If $S_{1},\ldots,S_{m}$ are Perron similarities with $S_{k}\in\mathsf{GL}_{n_{k}}(\mathbb{C})$ , then $\bigotimes_{k=1}^{m}S_{k}$ is a Perron similarity.

Proof.

Follows by a straightforward proof by induction on $m$ in conjunction with \threfkronPS. ∎

Theorem 7.16.

If $S_{1},\ldots,S_{m}$ are ideal with $S_{k}\in\mathsf{GL}_{n_{k}}(\mathbb{C})$ , then $\bigotimes_{k=1}^{m}S_{k}$ is ideal.

Proof.

Follows by a straightforward proof by induction on $m$ in conjunction with \threfkronideal. ∎

Corollary 7.17.

S\coloneqq\left(\bigotimes_{j=1}^{N}F_{n_{j}}\right)\otimes H_{2^{k}},\ k\in\mathbb{N}_{0},\ n_{j}\in\mathbb{N},\ j\in\left[N\right],

then $S$ is ideal and extremal.

Example 7.18.

The matrices $F_{24}$ , $H_{2}\otimes F_{12}$ , $H_{4}\otimes F_{6}$ , $H_{8}\otimes F_{3}$ , and $F_{4}\otimes F_{6}$ are ideal and extremal Perron similarities of order $24$ .

8. Geometrical representation of the spectra of 4-by-4 matrices

The problem of finding a geometric representation of all vectors $\begin{bmatrix}\lambda&\alpha&\omega\end{bmatrix}^{\top}$ in $\mathbb{R}^{3}$ such that $\{1,\lambda,\alpha+\omega\mathsf{i},\alpha-\omega\mathsf{i}\}$ is the spectrum of a 4-by-4 nonnegative matrix (we denote this region by $\mathbb{B}$ ) was posed by Egleston et al. [8, Problem 1].

In 2007, Torre-Mayo et al. [33] characterized the coefficients of the characteristic polynomials of four-by-four nonnegative matrices and in 2014, Benvenuti [2] used these results to produce the region given in Figure 1. It is worth noting here that this approach is not applicable to any other dimension.

Refer to caption — Figure 1. Geometrical representation of the spectra of four-by-four matrices by Benvenuti [2, Figure 11].

8.1. Region Generated by Spectratopes

Lemma 8.1.

S=\begin{bmatrix}1&e_{1}^{\top}\\ 0&F_{n}\end{bmatrix}\in\mathsf{GL}_{n+1}(\mathbb{C}),

then

S^{-1}=\begin{bmatrix}1&-e^{\top}/n\\ 0&F_{n}^{-1}\end{bmatrix}.

Proof.

Notice that

\displaystyle\begin{bmatrix}1&e_{1}^{\top}\\ 0&F_{n}\end{bmatrix}\begin{bmatrix}1&-e^{\top}/n\\ 0&F_{n}^{-1}\end{bmatrix}=\begin{bmatrix}1&-e^{\top}/n+e_{1}^{\top}F_{n}^{-1}\\ 0&I_{n}\end{bmatrix}

and since $F_{n}^{-1}=\frac{1}{n}\overline{F_{n}}$ , it follows that $e_{1}^{\top}F_{n}^{-1}=e^{\top}/n$ and the result follows. ∎

Theorem 8.2.

\thlabel

cartprodFn If

S=\begin{bmatrix}1&e_{1}^{\top}\\ 0&F_{n}\end{bmatrix},

then $\mathcal{P}(S)=[0,1]\times\mathcal{P}(F_{n})=[0,1]\times\mathcal{P}_{r}(F_{n})$ . Furthermore, $S$ is extremal.

Proof.

Let $x\in\mathbb{C}^{n+1}$ and let $y=\pi_{1}(x)\in\mathbb{C}^{n}$ . Since

e_{1}^{\top}D_{y}F_{n}^{-1}=\frac{y_{1}}{n}e^{\top},

it follows that

	$\displaystyle SD_{x}S^{-1}$	$\displaystyle=\begin{bmatrix}1&e_{1}^{\top}\\ 0&F_{n}\end{bmatrix}\begin{bmatrix}x_{1}&0\\ 0&D_{y}\end{bmatrix}\begin{bmatrix}1&-e^{\top}/n\\ 0&F_{n}^{-1}\end{bmatrix}$
		$\displaystyle=\begin{bmatrix}x_{1}&-\frac{x_{1}}{n}e^{\top}+e_{1}^{\top}D_{y}F_{n}^{-1}\\ 0&F_{n}D_{y}F_{n}^{-1}\end{bmatrix}$
		$\displaystyle=\begin{bmatrix}x_{1}&\frac{y_{1}-x_{1}}{n}e^{\top}\\ 0&F_{n}D_{y}F_{n}^{-1}\end{bmatrix}.$

If $x\in\mathcal{P}(S)$ , then the matrix above is stochastic. Thus, $x_{1}\in[0,1]$ and $y\in\mathcal{P}(F_{n})=\mathcal{P}_{r}(F_{n})$ , i.e., $x\in[0,1]\times\mathcal{P}_{r}(F_{n})$ .

If $x\in[0,1]\times\mathcal{P}_{r}(F_{n})$ , then $y\coloneqq\pi_{1}(x)\in\mathcal{P}_{r}(F_{n})=\mathcal{P}(F_{n})$ . Since the first column of $F_{n}e_{1}=e$ , it follows that $y_{1}=1$ and the matrix

SD_{x}S^{-1}=\begin{bmatrix}x_{1}&\frac{1-x_{1}}{n}e^{\top}\\ 0&F_{n}D_{y}F_{n}^{-1}\end{bmatrix}

is clearly stochastic, i.e., $x\in\mathcal{P}(S)$ .

Finally, note that $S$ is extremal because $F_{n}$ is extremal. ∎

If $S\in\mathsf{GL}_{4}(\mathbb{C})$ is a Perron similarity, then

S\sim\begin{bmatrix}1&1&1&1\\ 1&\lambda_{2}&\alpha_{2}+\omega_{2}\mathsf{i}&\alpha_{2}-\omega_{2}\mathsf{i}\\ 1&\lambda_{3}&\alpha_{3}+\omega_{3}\mathsf{i}&\alpha_{3}-\omega_{3}\mathsf{i}\\ 1&\lambda_{4}&\alpha_{4}+\omega_{4}\mathsf{i}&\alpha_{4}-\omega_{4}\mathsf{i}\end{bmatrix}.

Furthermore, if $S$ is ideal and $x\in\mathcal{P}_{r}(S)$ , then there are nonnegative scalars $\gamma_{1}$ , $\gamma_{2}$ , $\gamma_{3}$ , and $\gamma_{4}$ such that $\sum_{i=1}^{4}\gamma_{i}=1$ and

	$\displaystyle x$	$\displaystyle=\begin{bmatrix}1&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\lambda_{i}&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\left(\alpha_{i}+\omega_{i}\mathsf{i}\right)&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\left(\alpha_{4}-\omega_{4}\mathsf{i}\right)\end{bmatrix}$
		$\displaystyle=\begin{bmatrix}1&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\lambda_{i}&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\alpha_{i}+\sum_{i=2}^{4}\gamma_{i}\omega_{i}\mathsf{i}&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\alpha_{i}-\sum_{i=2}^{4}\gamma_{i}\omega_{i}\mathsf{i}\end{bmatrix}.$

Consequently,

\begin{bmatrix}\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\lambda_{i}&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\alpha_{i}&\sum_{i=2}^{4}\gamma_{i}\omega_{i}\end{bmatrix}\in\mathbb{B}

and

\begin{bmatrix}\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\lambda_{i}&\gamma_{1}+\sum_{i=2}^{4}\gamma_{i}\alpha_{i}&-\sum_{i=2}^{4}\gamma_{i}\omega_{i}\end{bmatrix}\in\mathbb{B}.

When $n=4$ , the K-arcs in the upper-half region are:

•

$K_{4}(0,1)$ (Type 0);
•

$K_{4}(1/4,1/3)$ (Type I); and
•

$K_{4}(1/3,1/2)$ (Type II).

However, it is known that $K_{4}(1/3,1/2)=\overline{K_{4}^{2}(1/4,1/3)}$ [27, Remark 5.3]. As mentioned earlier, the Type 0 arc is subsumed in the Type I arc. Thus, it suffices to consider Perron similarities of realizing matrices corresponding to the arc $K_{4}(1/4,1/3)$ . Figure 2 depicts the projected spectratope corresponding to $F_{4}$ .

Figure 4 contains spectra derived from the projected spectratopes of these Peron similarities. Notice that Figure 4 matches the Karpelevich region when $n=4$ .

If $S=\begin{bmatrix}1&e_{1}^{\top}\\ 0&F_{3}\end{bmatrix}\in\mathsf{GL}_{4}(\mathbb{C}),$ then $\mathcal{P}(S)=[0,1]\times\mathcal{P}_{r}(F_{3})$ by \threfcartprodFn. Figure 5 adds the projected spectratope of $S$ .

Notice that the missing region, which is small relative to the entire region, contains spectra such that $-1\leqslant\lambda\leqslant 0$ , $0\leqslant\alpha\leqslant 1$ , and $-1\leqslant\omega\leqslant 0$ .

9. Implications for Further Inquiry

Theorems LABEL:hadamardcones and LABEL:CSpolyconePSpolytope demonstrate that $\mathcal{C}(S)$ ( $\mathcal{P}(S)$ ) is a polyhedral cone (polytope) that is closed with respect to the Hadamard product. As such we pose the following.

Question 9.1.

If $K$ is a polyhedral cone (polytope) that is closed with respect to the Hadamard product, is there an invertible matrix $S$ such that $K=\mathcal{C}(S)$ ( $K=\mathcal{P}(S)$ )?

The following conjecture, which fails when $n=2$ and $n=3$ , would demonstrate that characterizing the extreme points is enough to characterize $\mathbb{SL}_{1}^{n}$ .

Conjecture 9.2.

If $n>3$ , then $\partial\mathbb{SL}_{1}^{n}\subseteq\mathbb{E}^{n}$ , i.e., points on the boundary are extremal.

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