Centrosymmetric nonnegative realization of spectra

TL;DR

This paper characterizes which spectra can be realized by centrosymmetric nonnegative matrices, providing conditions and specific cases where realizability is guaranteed, especially for real spectra and small matrix sizes.

Contribution

It introduces a characterization of spectra realizable by centrosymmetric nonnegative matrices, including new sufficient conditions and special cases for real spectra.

Findings

Lists of nonnegative real numbers are realizable by centrosymmetric matrices.

Lists of Suleimanova type (except one case) are realizable.

For n=4, all realizable real spectra are centrosymmetric realizable.

Abstract

A list $\Lambda =\{\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. In this paper we intent to characterize those lists of complex numbers, which are realizable by a centrosymmetric nonnegative matrix. In particular, we show that lists of nonnegative real numbers, and lists of complex numbers of Suleimanova type (except in one particular case), are always the spectrum of some centrosymmetric nonnegative matrix. For the general lists we give sufficient conditions via a perturbation result. We also show that for $n=4,$ every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix.

Equations

$$c_{i,j}=c_{m-i+1,n-j+1}.$$

$$C=\left[\begin{array}{cc}A& JBJ\\ B& JAJ\end{array}\right],$$

$$C=\left[\begin{array}{ccc}A& \mathbf{x}& JBJ\\ {\mathbf{y}}^T& c& {\mathbf{y}}^{T}J\\ B& J\mathbf{x}& JAJ\end{array}\right],$$

$$\left[\begin{array}{cc}A+JB& \\ & A-JB\end{array}\right].$$

$$\displaystyle\left.\begin{array}[]{c}\text{If }A\text{ has eigenvalues }\lambda_{1},\lambda_{2},\ldots,\lambda_{n},\text{ then }A+\mathbf{vq}^{T},\\ \text{has eigenvalues }\lambda_{1},\ldots,\lambda_{k-1},\lambda_{k}+\mathbf{v}^{T}\mathbf{q},\lambda_{k+1},\ldots,\lambda_{n},\\ \text{where }A\mathbf{v}=\lambda_{k}\mathbf{v}\text{ \ and \ }\mathbf{q\in\mathbb{C}}^{n}.\text{\ }\end{array}\right\}$$

$${\lambda }_i\in \mathcal{F}=\{{\lambda }_i\in \mathbb{C}:Re{\lambda }_i\le 0,|Re{\lambda }_i|\ge |Im{\lambda }_i|\},i=2,\dots ,n,$$

$$\Lambda =\{{\lambda }_1,{\lambda }_2,z_1,{\overline{z}}_1,\dots ,z_m,{\overline{z}}_m\},z_j=a_j+i{b}_j,a_j\in \mathbb{R},b_j>0,j=1,\dots ,m.$$

$${\Lambda }_k=\{{\lambda }_{k1},z_{k1},{\overline{z}}_{k1},\dots ,z_{k\frac{m}{2}},{\overline{z}}_{k\frac{m}{2}}\},k=1,2,{\lambda }_{11}={\lambda }_1,{\lambda }_{21}={\lambda }_2.$$

$$A+JB={\lambda }_1\underset{j=1}{\overset{\frac{m}{2}}{⨁}}\left[\begin{array}{cc}{a}_{1j}& -b_{1j}\\ b_{1j}& a_{1j}\end{array}\right]\text{and}A-JB={\lambda }_2\underset{j=1}{\overset{\frac{m}{2}}{⨁}}\left[\begin{array}{cc}{a}_{2j}& -b_{2j}\\ b_{2j}& a_{2j}\end{array}\right]$$

$$C=\left[\begin{array}{cc}A& JBJ\\ B& JAJ\end{array}\right]$$

$${\Lambda }_1$$

$${\Lambda }_2$$

$$A+JB={\lambda }_1\oplus {\lambda }_2\underset{j=1}{\overset{\lfloor \frac{m}{2}\rfloor }{⨁}}\left[\begin{array}{cc}{a}_{1j}& -b_{1j}\\ b_{1j}& a_{1j}\end{array}\right]\text{ and}$$

$$A-JB=\underset{j=1}{\overset{\lceil \frac{m}{2}\rceil }{⨁}}\left[\begin{array}{cc}{a}_{2j}& -b_{2j}\\ b_{2j}& a_{2j}\end{array}\right]$$

$$\Lambda =\{{\lambda }_1,z_1,{\overline{z}}_1,\dots ,z_m,{\overline{z}}_m\},z_j=a_j+i{b}_j,a_j\in \mathbb{R},b_j>0,j=1,\dots ,m.$$

$${\Lambda }_1$$

$${\Lambda }_2$$

$${\lambda }_1\underset{j=1}{\overset{\frac{m}{2}}{⨁}}\left[\begin{array}{cc}{a}_{1j}& -b_{1j}\\ b_{1j}& a_{1j}\end{array}\right]=\left[\begin{array}{cc}{\lambda }_1& \\ & A+JB\end{array}\right]\text{ and}$$

$$A-JB=\underset{j=1}{\overset{\frac{m}{2}}{⨁}}\left[\begin{array}{cc}{a}_{2j}& -b_{2j}\\ b_{2j}& a_{2j}\end{array}\right],$$

$$\left[\begin{array}{ccc}A& 0& JBJ\\ 0& {\lambda }_1& 0\\ B& 0& JAJ\end{array}\right]$$

$${\Lambda }_1$$

$${\Lambda }_2$$

$$\underset{j=1}{\overset{\lceil \frac{m}{2}\rceil }{⨁}}\left[\begin{array}{cc}{a}_{1j}& -b_{1j}\\ b_{1j}& a_{1j}\end{array}\right]=\left[\begin{array}{cccccc}{a}_{11}& -b_{11}& 0& 0& \cdots & 0\\ b_{11}& & & & & \\ 0& & & & & \\ 0& & & A+JB& & \\ ⋮& & & & & \\ 0& & & & & \end{array}\right]$$

$$A-JB={\lambda }_1\underset{j=1}{\overset{\lfloor \frac{m}{2}\rfloor }{⨁}}\left[\begin{array}{cc}{a}_{2j}& -b_{2j}\\ b_{2j}& a_{2j}\end{array}\right]$$

$$\left[\begin{array}{ccc}A& \mathbf{x}& JBJ\\ {\mathbf{y}}^T& a_{11}& {\mathbf{y}}^{T}J\\ B& J\mathbf{x}& JAJ\end{array}\right]$$

$${\Lambda }_1=\{{\lambda }_1,\dots ,{\lambda }_m\},{\Lambda }_2=\{{\lambda }_{m+1},\dots ,{\lambda }_n\}.$$

$$A=\frac{1}{2}diag\{{\lambda }_1+{\lambda }_{m+1},\dots ,{\lambda }_m+{\lambda }_n\},B=\frac{1}{2}Jdiag\{{\lambda }_1-{\lambda }_{m+1},\dots ,{\lambda }_m-{\lambda }_n\},$$

$$C=\left[\begin{array}{cc}A& JBJ\\ B& JAJ\end{array}\right].$$

$${\Lambda }_1=\{{\lambda }_1,\dots ,{\lambda }_{\lceil \frac{n}{2}\rceil }\},{\Lambda }_2=\{{\lambda }_{\lceil \frac{n}{2}\rceil +1},\dots ,{\lambda }_n\}.$$

$$\left[\begin{array}{cc}A+JB& \sqrt{2}\mathbf{x}\\ \sqrt{2}{\mathbf{y}}^T& c\end{array}\right]=diag\{{\lambda }_1,\dots ,{\lambda }_{\lceil \frac{n}{2}\rceil }\}$$

$$A$$

$$B$$

Full text

AMS classification:   15A18, 15A29

Key words: Inverse eigenvalue problem, nonnegative matrix, centrosymmetric matrix.

1 Introduction

The *nonnegative inverse eigenvalue problem *(hereafter NIEP) is the problem of finding necessary and suffcient conditions for the existence of a $n\times n$ entrywise nonnegative matrix with prescribed complex spectrum $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}.$ If there exists an $n\times n$ nonnegative matrix $A$ with spectrum $\Lambda$ we say that $\Lambda$ is realizable and that $A$ is the realizing matrix. A complete solution for the NIEP is known only for $n\leq 4,$ which shows the difficulty of the problem. Throughout this paper, if $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ is realizable by a nonnegative matrix $A$, then $\lambda_{1}=\rho(A)=\max\{\left|\lambda_{i}\right|,$ $\lambda_{i}\in\Lambda\}$ is the Perron eigenvalue of $A.$ We shall denote the transpose of a matrix $A$ by $A^{T}$, $\lfloor x\rfloor$ and $\lceil x\rceil$ denote the largest integer less than or equal to $x$ and the smallest integer greater than or equal to $x,$ respectively. $J$ will denote the counteridentity matrix, that is, $J=[\mathbf{e}_{n}\mid\mathbf{e}_{n-1}\mid\cdots\mid\mathbf{e}_{1}]$. Then it is clear that $J^{T}=J,J^{2}=I$.

Observe that multiplying a matrix $A$ by $J$ from the left results in reversing its rows, while multiplying $A$ by $J$ from the right results in reversing its columns. A vector $x$ is called symmetric if $Jx=x$.

The set of all $n\times n$ real matrices with constant row sums equal to $\alpha\in\mathbb{R}$ will be denote by $\mathcal{CS}_{\alpha}.$ It is clear that $\mathbf{e}=[1,1,\ldots,1]^{T}$ is an eigenvector of any matrix in $\mathcal{CS}_{\alpha},$ corresponding to the eigenvalue $\alpha.$ The relevance of matrices with constant row sums is due to the well known fact that the problem of finding a nonnegative matrix $A$ with spectrum $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ is equivalent to the problem of finding a nonnegative matrix $B\in\mathcal{CS}_{\lambda_{1}}$ with spectrum $\Lambda,$ that is, $A$ and $B$ are similar if the Perron eigenvalue is simple and they are cospectral otherwise (see [4]).

In this paper we study the NIEP for centrosymmetric matrices. Centrosymmetric matrices appear in many areas: physics, communication theory, differential equations, numerical analysis, engineering, statistics, etc. Now we state the definition and certain properties about centrosymmetric matrices.

Definition 1.1

A matrix $C=(c_{ij})\in M_{m,n}$ is said to be centrosymmetric, if its entries satisfy the relation

[TABLE]

The definition means that a centrosymmetric matrix $C$ can be written as $J_{m}CJ_{n}=C,$ where $J_{n}$ is the $n\times n$ counteridentity matrix. That is, $J_{n}=\left[\mathbf{e}_{n}\mid\mathbf{e}_{n-1}\mid\cdots\mid\mathbf{e}_{1}\right].$ For $m=n,$ we shall write $J$ instead $J_{n}.$

 

The following properties and results on centrosymmetric matrices are easy to verify (see [2]).

Lemma 1.1

[2*]** Let $C_{1}$ and $C_{2}$ be centrosymmetric matrices, and $\alpha_{1},\alpha_{2}\in\mathbb{R}.$ Then,

$i)$ $C_{1}^{-1}$, if exists, $ii)$ $C_{1}^{T}$, $iii)$ $\alpha_{1}C_{1}\pm\alpha_{2}C_{2}$, $iv)$ $C_{1}C_{2}$ are all centrosymmetric.*

Lemma 1.2

[2]** Let $C$ be an $n\times n$ centrosymmetric nonnegative matrix. If $\mathbf{v}$ is an eigenvector of $C$ corresponding to the eigenvalue $\lambda,$ then $J\mathbf{v}$ is also an eigenvector of $C$ corresponding to $\lambda$. Moreover, if $\lambda_{1}$ is the Perron eigenvalue of $C$, then there is a nonnegative eigenvector $\mathbf{x}$ such that $J\mathbf{x}=\mathbf{x}$.

Theorem 1.1

[2*]** Let $C$ be an $n\times n$ centrosymmetric matrix.

$i)$ If $n=2m,$ then $C$ can be written as*

[TABLE]

*where $A,B$ and $J$ are $m\times m$ matrices.

$ii)$ If $n=2m+1,$ then $C$ can be written as*

[TABLE]

where $A,B$ and $J$ are $m\times m$ matrices, $\mathbf{x}$ and $\mathbf{y}$ are $m$-dimensional vectors, and $c$ is a real number.

Theorem 1.2

[2*]** Let $C$ be an $n\times n$ centrosymmetric matrix.

$i)$ If $C=\begin{bmatrix}A&JBJ\\ B&JAJ\end{bmatrix}$, then $C$ is orthogonally similar to the matrix*

[TABLE]

*Moreover, if $C$ is nonnegative with the Perron eigenvalue $\lambda_{1}$, then $\lambda_{1}$ is the Perron eigenvalue of $A+JB$.

$ii)$ If $C$ is written as $C=\begin{bmatrix}A&\mathbf{x}&JBJ\\ \mathbf{y}^{T}&c&\mathbf{y}^{T}J\\ B&J\mathbf{x}&JAJ\end{bmatrix}$, then $C$ is orthogonally similar to the matrices $\begin{bmatrix}c&\sqrt{2}\mathbf{y}^{T}&\\ \sqrt{2}\mathbf{x}&A+JB&\\ &&A-JB\end{bmatrix}$ and

$\begin{bmatrix}A+JB&\sqrt{2}\mathbf{x}&\\ \sqrt{2}\mathbf{y}^{T}&c&\\ &&A-JB\end{bmatrix}$. Moreover, if $C$ is a nonnegative matrix with the Perron eigenvalue $\lambda_{1}$, then $\lambda_{1}$ is the Perron eigenvalue of $\begin{bmatrix}c&\sqrt{2}\mathbf{y}^{T}\\ \sqrt{2}\mathbf{x}&A+JB\end{bmatrix}$.*

The following theorem, due to R. Rado and published by H. Perfect in [6], will be used to throughout the paper to show some of our results. Rado’s theorem show how to modify $r$ eigenvalues of an $n\times n$ matrix, $r<n,$ via a rank-$r$ perturbation, without changing any of the remaining $n-r$ eigenvalues (see [3, 9] for the way in which Rado’s result has been applied to the NIEP). The case $r=1,$ is the well known Brauer’s theorem [1], which states that

[TABLE]

Theorem 1.3

Rado [6] Let $M$ be an $n\times n$ arbitrary matrix with spectrum $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}.$ Let $X=\left[\mathbf{x}_{1}\mid\cdots\mid\mathbf{x}_{r}\right]$ be such that $rank(X)=r$ and $M\mathbf{x}_{i}=\lambda_{i}\mathbf{x}_{i},$ $i=1,\ldots,r,$ $r<n.$ Let $\mathcal{C}$ be an $r\times n$ arbitrary matrix. Then $M+X\mathcal{C}$ has eigenvalues $\mu_{1},\ldots,\mu_{r},\lambda_{r+1},\ldots,\lambda_{n},$ where $\mu_{1},\ldots,\mu_{r}$ are eigenvalues of the matrix $\Omega+\mathcal{C}X$ with $\Omega=diag\{\lambda_{1},\ldots,\lambda_{r}\}.$

In [10] the authors prove that a realizable list of complex numbers of Suleimanova type $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\},$ that is, with $\lambda_{1}$ being the Perron eigenvalue, and

[TABLE]

is in particular realizable with spectrum $\Lambda$ and with prescribed diagonal entries $\omega_{1},\omega_{2},\ldots,\omega_{n}$ if and only if $\sum\limits_{i=1}^{n}\omega_{i}=\sum\limits_{i=1}^{n}\lambda_{i}.$ This result, which we shall use in Section $3,$ follows directly from the following lemma and (4), which we set here for sake of completeness:

Lemma 1.3

Let $\Lambda^{\prime}=\{-\sum\limits_{i=2}^{n}\lambda_{i},\lambda_{2},\ldots,\lambda_{n}\}$ be a realizable list of complex numbers, with Perron eigenvalue $-\sum\limits_{i=2}^{n}\lambda_{i}.$ Then for any $\lambda_{1}\geq-\sum\limits_{i=2}^{n}\lambda_{i},$ the list $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ is the spectrum of a nonnegative matrix with prescribed diagonal entries $\Gamma=\{\omega_{1},\omega_{2},\ldots,\omega_{n}\},$ if and only if $\sum\limits_{i=1}^{n}\omega_{i}=\sum\limits_{i=1}^{n}\lambda_{i}.$

Proof. Since $\Lambda^{\prime}$ is realizable, there exists a nonnegative matrix $B\in\mathcal{CS}_{\beta},$ where $\beta=-\sum\limits_{i=2}^{n}\lambda_{i},$ with spectrum $\Lambda^{\prime}$ and $tr(B)=0.$ Let $\mathbf{q}^{T}=\left[\omega_{1},\ldots,\omega_{n}\right].$ Then from (4), $A=B+\mathbf{eq}^{T}$ is a nonnegative matrix with spectrum $\Lambda$ and with diagonal entries $\omega_{1},\omega_{2},\ldots,\omega_{n}$.

2 Real centrosymmetric matrices with prescribed spectrum

In this section we show that a list of complex numbers $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\},$ with $\Lambda=\overline{\Lambda}$, is always the spectrum of a real centrosymmetric matrix (not necessarily nonnegative). That is, the real centrosymmetric inverse eigenvalue problem for a list of conjugate complex numbers has always a solution.

Theorem 2.1

Let $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ be a list of complex numbers, with $\Lambda=\overline{\Lambda}$, $n\geq 3$. Then $\Lambda$ is the spectrum of an $n\times n$ real centrosymmetric matrix.

Proof. We shall distinguish two cases:

Case 1. Let $n$ be even. First we consider $\Lambda$ with only two real numbers, that is,

[TABLE]

If $m$ is even, we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ with

[TABLE]

Then, from Theorem 1.2 the $(m+1)\times(m+1)$ matrices

[TABLE]

have spectrum $\Lambda_{1}$ and $\Lambda_{2}$ respectively. Then $A=\frac{1}{2}(A+JB+(A-JB))$ and $B=\frac{1}{2}J(A+JB-(A-JB))$. Therefore

[TABLE]

is real centrosymmetric with spectrum $\Lambda$.

If $m$ is odd, we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ with

[TABLE]

Then,

[TABLE]

[TABLE]

have the spectra $\Lambda_{1}$ and $\Lambda_{2},$ respectively and the proof follows as above. It is clear that if $\Lambda$ has $r$ real numbers, $r,m$ even, we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ as above, with $\Lambda_{k},$ $k=1,2,$ having $m+\frac{r}{2}$ numbers, $m$ complex numbers and $\frac{r}{2}$ real numbers. If $m$ is odd and $\Lambda$ has $r$ real numbers, $r$ even, then $\Lambda_{1}$ will have $\lfloor\frac{m}{2}\rfloor$ complex numbers plus $\frac{r+2}{2}$ real numbers, while $\Lambda_{2}$ will have $\lceil\frac{m}{2}\rceil$ complex numbers plus $\frac{r-2}{2}$ real numbers. Then the proof follows as before.

 

Case 2: Let $n$ be odd. First we consider $\Lambda$ with only one real number, that is,

[TABLE]

If $m$ is even we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ with

[TABLE]

Then the matrices

[TABLE]

[TABLE]

have spectrum $\Lambda_{1}$ and $\Lambda_{2},$ respectively. Then $A=\frac{1}{2}(A+JB+(A-JB))$ and $B=\frac{1}{2}J(A+JB-(A-JB))$. Hence from Theorem 1.2

[TABLE]

is real centrosymmetric with spectrum $\Lambda$.

On the other hand, if $m$ is odd, a partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ must be of the form

[TABLE]

Then the matrices

[TABLE]

and

[TABLE]

have spectrum $\Lambda_{1}$ and $\Lambda_{2}$ respectively. Then $A=\frac{1}{2}(A+JB+(A-JB))$ and $B=\frac{1}{2}J(A+JB-(A-JB))$. Hence

[TABLE]

where $\mathbf{x}=\frac{1}{\sqrt{2}}[b_{11},0\cdots,0]^{T}$ and $\mathbf{y}^{T}=\frac{1}{\sqrt{2}}[-b_{11},\cdots,0]$, is a real centrosymmetric matrix with spectrum $\Lambda$.

It is easy to see that the list $\Lambda=\{\lambda_{1},z_{1},\overline{z}_{1},\ldots,z_{m},\overline{z}_{m}\}$ can be extended to a list $\Lambda^{\prime}=\{\lambda_{1},\ldots,\lambda_{p},z_{1},\overline{z}_{1},\ldots,z_{m},\overline{z}_{m}\}$ with $p$ real numbers, $p$ odd, which always admit a partition in two self-conjugate lists $\Lambda_{1}$, $\Lambda_{2}$ such that $\Lambda_{1}$ has $\lceil\frac{2m+p}{2}\rceil$ eigenvalues and $\Lambda_{2}$ has the remaining eigenvalues, and a real centrosymmetric matrix with spectrum $\Lambda^{\prime}$ can be obtained as above.

3 Centrosymmetric nonnegative inverse eigenvalue problem

In this section we study the NIEP for centrosymmetric matrices. First, we show that lists of real nonnegative numbers are always realizable for a centrosymmetric nonnegative matrix. Second, we show that lists of complex numbers of Suleimanova type [11] are also realizable for centrosymmetric nonnegative matrices, except if the list has only one real number and $m$ pairs of conjugated complex numbers, with $m$ being odd. Third, for the general lists, we give sufficient conditions for the existence of a centrosymmetric nonnegative matrix with prescribed complex spectrum via a perturbation result. Finally, we study the centrosymmetric realizability of lists of complex numbers of size $n=4$ with prescribed diagonal entries.

We start by showing that a list of real nonnegative numbers $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ is always realizable by a centrosymmetric matrix. In addition, if $\lambda_{1}$ is simple, $\Lambda$ is realizable by a centrosymmetric positive matrix.

Theorem 3.1

Let $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ be a list of nonnegative real numbers with $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0$. Then $\Lambda$ is realizable by an $n\times n$ centrosymmetric matrix.

Proof. For even $n=2m,$ we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, where

[TABLE]

Then for $A+JB=diag\{\lambda_{1},\ldots,\lambda_{m}\}$ and $A-JB=diag\{\lambda_{m+1},\ldots,\lambda_{n}\}$ we have that

[TABLE]

are nonnegative matrices, and a solution matrix is of the form

[TABLE]

For odd $n=2m+1,$ we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, where

[TABLE]

Then for

[TABLE]

and $A-JB=diag\{\lambda_{\lceil\frac{n}{2}\rceil+1},\ldots,\lambda_{n}\},$ we have $A+JB=diag\{\lambda_{1},\ldots,\lambda_{\lfloor\frac{n}{2}\rfloor}\}$, $c=\lambda_{\lceil\frac{n}{2}\rceil}$. Therefore

[TABLE]

are nonnegative matrices and a solution matrix is

[TABLE]

In [5] Perfect introduces the $n\times n$ matrix

[TABLE]

and she proves that if $D=diag\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\},$ with $\lambda_{1}>\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0,$ then $A=PDP^{-1}$ is a positive matrix in $\mathcal{CS}_{\lambda_{1}}.$ As a consequence, we have the following result, which gives a very simple way to compute a centrosymmetric positive matrix with prescribed nonnegative spectrum.

Theorem 3.2

Let $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ be a list of nonnegative real numbers with $\lambda_{1}>\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0$. Then $\Lambda$ is realizable by an $n\times n$ centrosymmetric positive matrix.

Proof. For even $n=2m,$ we take the partition  $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, with

[TABLE]

We set $A+JB=PDP^{-1}>0$, where $D=diag\{\lambda_{1},\ldots,\lambda_{m}\},$ $P$ is the Perfect matrix in (5), and $A-JB=diag\{\lambda_{m+1},\ldots,\lambda_{n}\}$. Then

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

is a centrosymmetric matrix with spectrum $\Lambda$. It remains to prove that $C$ is positive. It is enough to show that $B$ is positive. In fact, if $d_{jj}$, $j=1,2,\ldots,m,$ are the diagonal entries of $PDP^{-1},$ we must show that

[TABLE]

It is easy to see that the diagonal entries of the matrix $PDP^{-1}$ are given by:

[TABLE]

For $j=1,$ $d_{11}>\lambda_{m+1}.$ In fact: since

[TABLE]

then by adding the inequalities we have

[TABLE]

Since $d_{11}=d_{22}$ and $\lambda_{m+1}\geq\lambda_{m+2}$, then $d_{22}>\lambda_{m+2}$.

By proceeding in the same way we have $d_{jj}>\lambda_{m+j},$ $j=3,4,\ldots,m$. Therefore $B>0$, and the matrix $C$ in (6) is centrosymmetric positive.

If $n=2m+1$ is odd, then we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ with

[TABLE]

Then for $\begin{bmatrix}A+JB&\sqrt{2}\mathbf{x}\\ \sqrt{2}\mathbf{y}^{T}&c\end{bmatrix}=PDP^{-1}=\begin{bmatrix}A_{11}&\mathbf{a}\\ \mathbf{b}^{T}&a_{m+1,m+1}\end{bmatrix}>0$, and

$A-JB=diag\{\lambda_{\lceil\frac{n}{2}\rceil+1},\ldots,\lambda_{n}\},$ we have

[TABLE]

[TABLE]

$B$ is positive from the same argument as in the even case. Therefore the matrix

[TABLE]

is centrosymmetric positive with spectrum $\Lambda.$    

Next we shall see an anomalous case. That is, we shall prove that if a list realizable $\Lambda$ has only one real positive number and $m$ pairs of conjugated complex numbers, with $m$ being odd, then $\Lambda$ cannot be the spectrum of a centrosymmetric nonnegative matrix.

Theorem 3.3

Let $\Lambda=\{\lambda_{1},z_{1},\overline{z}_{1},\ldots,z_{m},\overline{z}_{m}\}$ be a realizable list of complex numbers, with odd $m$, $z_{j}=a_{j}+ib_{j}$ $j=1,\ldots,m$, $a\in\mathbb{R}$, $b_{j}>0$. Then $\Lambda$ cannot be the spectrum of a centrosymmetric nonnegative matrix.

Proof. Suppose that $\Lambda$ is the spectrum of a centrosymmetric nonnegative matrix $C$ of order $2m+1.$ Then $C$ is of the form

[TABLE]

where $A$ and $B$ are $m\times m$ nonnegative matrices, $\mathbf{x}$, $\mathbf{y}$ are nonnegative $m\times 1$ matrices, and $c$ is a nonnegative real number. From Theorem 1.2 $C$ is orthogonally similar to the matrix

[TABLE]

where $\begin{bmatrix}c&\sqrt{2}\mathbf{y}^{T}\\ \sqrt{2}\mathbf{x}&A+JB\end{bmatrix}$ is an $(m+1)\times(m+1)$ nonnegative matrix with $m+1$ eigenvalues of $C$ including the Perron eigenvalue. That is, $C$ has $m$ (an odd number) complex eigenvalues, which is a contradiction.    

 

Now we consider the centrosymmetric realizability of lists of Suleimanova type. We start with the following simple result:

Lemma 3.1

Let $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ be the spectrum of a centrosymmetric nonnegative matrix and let $\epsilon>0$. Then $\{\lambda_{1}+\epsilon,\lambda_{2},\ldots,\lambda_{n}\}$ is also the spectrum of a centrosymmetric nonnegative matrix.

Proof. Let $C$ be a centrosymmetric nonnegative matrix with spectrum $\Lambda$. Then from Lemma 1.2, there exists $\mathbf{v}\geq 0$, $C\mathbf{v}=\lambda_{1}\mathbf{v}$, with $J\mathbf{v}=\mathbf{v,}$ and from Brauer’s Theorem (see (4)) the matrix

[TABLE]

is centrosymmetric nonnegative with spectrum $\{\lambda_{1}+\epsilon,\lambda_{2},\ldots,\lambda_{n}\}$.    

It is well known that lists of complex numbers $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ of Suleimanova type are realizable by a nonnegative matrix if only if $\sum_{i=1}^{n}\lambda_{i}\geq 0$. We prove that a list of this type is in particular realizable by a centrosymmetric nonnegative matrix, except if the list is as in the Theorem 3.3, that is, it has only one real positive number and $m$ pairs of conjugated complex numbers, with $m$ being odd.

The following result show that any realizable list of complex numbers of Suleimanova type, with two real numbers and $m$ pairs of complex numbers is always the spectrum of a centrosymmetric nonnegative matrix of order $n=2m+2.$ As a consequence, all realizable lists of complex numbers of Suleimanova type, with an even number of elements, are always realizable by centrosymmetric matrices.

Lemma 3.2

Let $\Lambda=\{\lambda_{1},\lambda_{2},z_{1},\overline{z}_{1},\ldots,z_{m},\overline{z}_{m}\}$, with $\lambda_{2},z_{j},\overline{z}_{j}\in\mathcal{F}$, $z_{j}=a_{j}+ib_{j}$, $a_{j}\in\mathbb{R}$, $b_{j}>0$, $j=1,\ldots,m,$ be a realizable list of complex numbers. Then $\Lambda$ is the spectrum of a centrosymmetric nonnegative matrix.

Proof. Since $\Lambda$ is realizable if only if $\sum_{i=1}^{n}\lambda_{i}\geq 0$, we take the list

[TABLE]

For even $m$ we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ with

[TABLE]

Then,

[TABLE]

has spectrum $\Lambda_{2}$, and from Lemma 1.3 we can always compute a nonnegative matrix $A+JB$ with spectrum $\Lambda_{1}$ and diagonal entries

[TABLE]

Since $\left|Rez_{j}\right|\geq\left|Imz_{j}\right|,$ $j=1,\ldots,m,$ both matrices,

[TABLE]

are nonnegative and

[TABLE]

is centrosymmetric nonnegative with spectrum $\Lambda^{\prime}$. Finally, from Lemma 3.1, $C=C^{\prime}+\frac{\sum_{i=1}^{n}\lambda_{i}}{\mathbf{v}^{T}\mathbf{v}}\mathbf{v}\mathbf{v}^{T}$, where $\mathbf{v}$ is the Perron eigenvector of $C^{\prime}$, is centrosymmetric nonnegative with spectrum $\Lambda$.

For odd $m$ we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ with

[TABLE]

and the proof follows as above.    

More generally, if $\Lambda$ has $r\geq 2$ real numbers, with even $r$, we have the following result:

Corollary 3.1

Let $\Lambda=\{\lambda_{1},\ldots,\lambda_{r},z_{1},\overline{z}_{1},\ldots,z_{m},\overline{z}_{m}\}$, with $z_{j},\overline{z}_{j}\in\mathcal{F}$, $z_{j}=a_{j}+ib_{j}$, $a_{j}\in\mathbb{R}$, $b_{j}>0$, $j=1,\ldots,m$; $\lambda_{j}\in\mathcal{F}$, $j=2,\ldots,r$, be a realizable list of complex numbers. Then $\Lambda$ is the spectrum of a centrosymmetric nonnegative matrix.

Proof. We consider the list

[TABLE]

Then, for even $m$ we take the partition of $\Lambda^{\prime}=\Lambda_{1}\cup\Lambda_{2}$ with:

[TABLE]

while for odd $m$ we take

[TABLE]

and the proof follows as the proof of Lemma 3.2.    

 

Now, we consider lists of complex numbers of Suleimanova type with an odd number of elements. We start with the following result:

Lemma 3.3

Let $\Lambda=\{\lambda_{1},z_{1},\overline{z}_{1},\ldots,z_{m},\overline{z}_{m}\}$ be a list of complex numbers with $z_{j},\overline{z}_{j}\in\mathcal{F}$, $z_{j}=a_{j}+ib_{j}$, $a_{j}\in\mathbb{R}$, $b_{j}>0$, $j=1,\ldots,m,$ $m$ even, which is realizable. Then $\Lambda$ is the spectrum of a centrosymmetric nonnegative matrix.

Proof. We consider the list

[TABLE]

with the partition $\Lambda^{\prime}=\Lambda_{1}\cup\Lambda_{2}$, where

[TABLE]

Then

[TABLE]

is an $m\times m$ real matrix with spectrum $\Lambda_{2},$ and

[TABLE]

of order $m+1,$ has spectrum $\Lambda_{1}$ and diagonal entries

[TABLE]

which there exists from Lemma 1.3. Then $A=\frac{1}{2}(A_{22}+A-JB)$ and $B=\frac{1}{2}J(A_{22}-(A-JB))$, are both nonnegative and the matrix

[TABLE]

is centrosymmetric nonnegative with spectrum $\Lambda^{\prime}$. Thus, $C=C^{\prime}+\frac{\sum_{i=1}^{n}\lambda_{i}}{\mathbf{v}^{T}\mathbf{v}}\mathbf{v}\mathbf{v}^{T}$, where $\mathbf{v}$ is the Perron eigenvector of $C^{\prime}$, is centrosymmetric nonnegative with spectrum $\Lambda$.   

More generally, if $\Lambda$ has $p$ real numbers, with odd $p,$ we have the following result:

Corollary 3.2

Let $\Lambda=\{\lambda_{1},\ldots,\lambda_{p},z_{1},\overline{z}_{1},\ldots,z_{m},\overline{z}_{m}\}$, with $z_{j},\overline{z}_{j}\in\mathcal{F}$, $z_{j}=a_{j}+ib_{j}$, $a_{j}\in\mathbb{R}$, $b_{j}>0$, $j=1,\ldots,m$; $\lambda_{k}\in\mathcal{F},$ $k=2,\ldots,p,$ be a realizable list of complex numbers. Then $\Lambda$ is the spectrum of a centrosymmetric nonnegative matrix.

Proof. Case even $m$. We consider the list

[TABLE]

and take the partition $\Lambda^{\prime}=\Lambda_{1}\cup\Lambda_{2}$ with,

[TABLE]

Then

[TABLE]

is a real matrix with spectrum $\Lambda_{2}$, and there exists a nonnegative matrix $A+JB$ with spectrum $\Lambda_{1}$ and diagonal entries

[TABLE]

Then the proof follows as before.

 

Case odd $m$: We consider a partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, in which the list $\Lambda_{1}$ has the Perron eigenvalue and only one more element than the list $\Lambda_{2}.$ Both lists, $\Lambda_{1}$ and $\Lambda_{2}$ must be self-conjugated. Partitions of this type always exist if $p\geq 3.$ The construction of a nonnegative centrosymmetric matrix follows as in the proof of Lemma 3.3.

Remark 3.1

According to the results in this section, it is clear that a realizable list of real numbers of Suleimanova type is in particular realizable by a centrosymmetric nonnegative matrix.

We conclude this section by presenting a sufficient condition for the existence and construction of centrosymmetric nonnegative matrices with prescribed spectrum**.**

Let $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ be a realizable list of complex numbers. In order to construct a centrosymmetric nonnegative matrix $A$ with spectrum $\Lambda$ we consider the following partition of $\Lambda:$

[TABLE]

where some of the lists $\Lambda_{k}$ can be empty. For each list $\Lambda_{k},$ $k\neq\lceil\frac{p_{0}}{2}\rceil,$ we associate the list

[TABLE]

which is realizable by a $(p_{k}+1)\times(p_{k}+1)$ nonnegative matrix $A_{k}.$ In particular, $A_{k}$ can be chosen as $A_{k}\in\mathcal{CS}_{\omega_{k}}.$ To the sub-list $\Lambda_{\lceil\frac{p_{0}}{2}\rceil}$ we associate the list $\Gamma_{\lceil\frac{p_{0}}{2}\rceil},$ which is the spectrum of a centrosymmetric nonnegative matrix $A_{\lceil\frac{p_{0}}{2}\rceil}$ of order $p_{\lceil\frac{p_{0}}{2}\rceil+1}.$ Then, the block diagonal matrices

[TABLE]

are centrosymmetric nonnegative with spectrum

[TABLE]

respectively. Of course, the matrices $J$ in (9) have the same order $p_{k}+1$ of the corresponding matrices $A_{k}$, $k=1,2,,\ldots,\lfloor\frac{p_{0}}{2}\rfloor$.

Let $X$ be an $n\times p_{0}$ matrix whose columns are nonnegative eigenvectors of the matrix $A$ in (9), that is,

[TABLE]

and

[TABLE]

where, $\mathbf{x}_{k}=\mathbf{e}^{T}=[1,1,\ldots,1]$ is the Perron eigenvector of $A_{k}$ in (9), $k=1,2,\ldots,\frac{p_{0}}{2}$ $(\lfloor\frac{p_{0}}{2}\rfloor),$ and $\mathbf{y}_{\lceil\frac{p_{0}}{2}\rceil}$ is the Perron eigenvector of $A_{\lceil\frac{p_{0}}{2}\rceil}$, with $\mathbf{y}_{\lceil\frac{p_{0}}{2}\rceil}=J\mathbf{y}_{\lceil\frac{p_{0}}{2}\rceil}\geq 0$ (symmetric vector). Observe that

[TABLE]

Now we state the main result of this section. First, we consider the even $p_{0}$ case:

Theorem 3.4

Let $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ be a list of complex numbers with $\Lambda=\overline{\Lambda},$ $\lambda_{1}\geq\left|\lambda_{i}\right|,$ $i=2,3,\ldots,n,$ and $\sum\limits_{i=1}^{n}\lambda_{i}\geq 0.$ Suppose there exists a partition of $\Lambda$ (as defined in (7)),

[TABLE]

*where some of the lists $\Lambda_{k}$ can be empty, such that the following conditions are satisfied:

$i)$ For each $k=1,2,\ldots,\frac{p_{0}}{2},$ there exists a nonnegative matrix with spectrum*

[TABLE]

$ii)$* There exists a centrosymmetric nonnegative matrix of order $p_{0},$ with spectrum $\Lambda_{0}$ and diagonal entries $\omega_{1},\omega_{2},\ldots,\omega_{\frac{p_{0}}{2}},\omega_{\frac{p_{0}}{2}},\ldots,\omega_{2},\omega_{1}.$

Then $\Lambda$ is realizable by an $n\times n$ centrosymmetric matrix.*

Proof. From $i)$ let $A_{k}$ be a $(p_{k}+1)\times(p_{k}+1)$ nonnegative matrix with spectrum $\Gamma_{k},$ $k=1,2,\ldots,\frac{p_{0}}{2}.$ We may assume that $A_{k}\in\mathcal{CS}_{\omega_{k}}.$ Then $A_{k}\mathbf{e}=\omega_{k}\mathbf{e},$ The matrix

[TABLE]

is centrosymmetric nonnegative with spectrum

[TABLE]

Let $X$ be the matrix in (10). Then $X$ is centrosymmetric nonnegative of zeros and ones.

Now, from $ii)$ and Theorem 1.3  let $B=\Omega+\mathcal{C}X$ centrosymmetric nonnegative matrix of order $p_{0}$ with spectrum $\Lambda_{0}$ and diagonal entries

[TABLE]

Then $\mathcal{C}X=B-\Omega=\begin{bmatrix}b_{1}&\cdots&b_{\frac{p_{0}}{2}}&J_{p_{0}}b_{\frac{p_{0}}{2}}&\cdots&J_{p_{0}}b_{1}\end{bmatrix}$ is centrosymmetric nonnegative, where $b_{k}$ and $J_{p_{0}}b_{k}$, $k=1,\ldots,\frac{p_{0}}{2}$, are the columns of $B-\Omega$. Therefore the $p_{0}\times n$ matrix $\mathcal{C}$ can be obtained as follows

[TABLE]

It is easy to see that $\mathcal{C}$ is centrosymmetric nonnegative. Then the perturbation $A+X\mathcal{C}$, with $A,X,\mathcal{C}$ as in (9),(10),(12) respectively, is nonnegative centrosymmetric and by Theorem 1.3, $A+X\mathcal{C}$ have the spectrum $\Lambda$.    

Now we consider the odd $p_{0}$ case:

Theorem 3.5

Let $\Lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$ be a list of complex numbers with $\Lambda=\overline{\Lambda},$ $\lambda_{1}\geq\left|\lambda_{i}\right|,$ $i=2,3,\ldots,n,$ and $\sum\limits_{i=1}^{n}\lambda_{i}\geq 0.$ Suppose there exists a partition of $\Lambda$ (as defined in (7)),

[TABLE]

*where some of the lists $\Lambda_{k}$ can be empty, such that the following conditions are satisfied:

$i)$ For each $k=1,2,\ldots,\lfloor\frac{p_{0}}{2}\rfloor,$ there exists a nonnegative matrix with spectrum*

[TABLE]

*while for $k=\lceil\frac{p_{0}}{2}\rceil$ there exists a centrosymmetric nonnegative matrix with spectrum $\Gamma_{\lceil\frac{p_{0}}{2}\rceil}$.

$ii)$ There exists a centrosymmetric nonnegative matrix of order $p_{0},$ with spectrum $\Lambda_{0}$ and diagonal entries $\omega_{1},\omega_{2},\ldots,\omega_{\lfloor\frac{p_{0}}{2}\rfloor},\omega_{\lceil\frac{p_{0}}{2}\rceil},\omega_{\lfloor\frac{p_{0}}{2}\rfloor},\ldots,\omega_{2},\omega_{1}.$

Then $\Lambda$ is realizable by an $n\times n$ centrosymmetric matrix.*

Proof. Case 1. The list $\Gamma_{\lceil\frac{p_{0}}{2}\rceil}$ is realizable by a centrosymmetric nonnegative matrix of odd order: In this case the proof follows as the proof of Theorem 3.4, with a perturbation of the form $A+X\mathcal{C}$, with $A,X$ as in (9), (11) respectively, and

[TABLE]

Case 2. The list $\Gamma_{\lceil\frac{p_{0}}{2}\rceil}$ is realizable by a centrosymmetric nonnegative matrix of order even: In this case we consider $A$ as in (9), $X$ as in (11) with all its columns normalized, and in $ii)$ we take $B=\Omega+\mathcal{C}$. Then $\mathcal{C}=B-\Omega$, is centrosymmetric nonnegative, and by Lemma 1.1 and Theorem 1.3, the perturbation $A+X\mathcal{C}X^{T}$ is centrosymmetric nonnegative with spectrum $\Lambda$.    

To apply Theorem 3.4 and Theorem 3.5, we need to know conditions under which there exists a $p_{0}\times p_{0}$ centrosymmetric nonnegative matrix with spectrum $\Lambda_{0}$ and diagonal entries

[TABLE]

In [8, Theorem $3.1$] (see also [7]) it has been proved that the spectrum of a $3\times 3$ nonnegative centrosymmetric matrix is real. Then, since bisymmetric matrices are centrosymmetric, we may use the necessary and sufficient conditions given in [3], for the existence of a $3\times 3$ centrosymmetric nonnegative matrix with real eigenvalues $\lambda_{1},\lambda_{2},\lambda_{3}$ and diagonal entries $\omega_{1},\omega_{2},\omega_{1}.$ For $n=4$ we have, in the Theorems 3.7 and 3.8 below, sufficient conditions for the existence and construction of a centrosymmetric nonnegative matrix with prescribed eigenvalues and diagonal entries.

The following result show that for lists of four real numbers, both problems, the nonnegative inverse eigenvalue problem and the centrosymmetric nonnegative inverse eigenvalue problem, are equivalent.

Theorem 3.6

Let $\Lambda=\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\}$ be a list realizable of real numbers. Then $\Lambda$ is in particular realizable by a centrosymmetric nonnegative matrix.

Proof. Case 1: If $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}\geq 0,$ the result follows from Theorem 3.1 when $n=4$.

Case 2: If $\lambda_{1}>0\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$, that is, if the list is of real Suleimanova type, then the result follows from Remark 3.1.

Case 3: If $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq 0>\lambda_{4}$, we have the following general result:

A list realizable of real numbers with only one negative eigenvalue is also realizable by a centrosymmetric nonnegative matrix. In fact, let $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$, with $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n-1}\geq 0>\lambda_{n}$.

If $n=2m$ is even we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, with

[TABLE]

and we set

[TABLE]

and the proof follows as before.

If $n=2m+1$ is odd we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, with

[TABLE]

and we set

[TABLE]

[TABLE]

Then, the proof follows as before.

Case 4: If $\lambda_{1}\geq\lambda_{2}\geq 0>\lambda_{3}\geq\lambda_{4}$, with $\lambda_{2}+\lambda_{3}>0$;

 we take the partition $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, with $\Lambda_{1}=\{\lambda_{1},\lambda_{2}\}$, $\Lambda_{2}=\{\lambda_{3},\lambda_{4}\},$ and we set

[TABLE]

Then, the proof follows as before.

If $\lambda_{1}\geq\lambda_{2}\geq 0>\lambda_{3}\geq\lambda_{4}$, with $\lambda_{2}+\lambda_{3}<0;$

we take the partition $\Lambda_{1}=\{\lambda_{1},\lambda_{2}\},\ \ \Lambda_{2}=\{\lambda_{3},\lambda_{4}\}$ and we set

[TABLE]

[TABLE]

Then

[TABLE]

[TABLE]

are both nonnegative matrices, and

[TABLE]

is a $4\times 4$ centrosymmetric nonnegative matrix.    

Now we give a sufficient condition for the existence and construction of a $4\times 4$ centrosymmetric nonnegative matrix with prescribed eigenvalues and diagonal entries. We start with lists of real numbers.

Theorem 3.7

*Let $\Lambda=\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\}$ be a list of real numbers with $\sum_{i=1}^{4}\lambda_{i}\geq 0$, $\lambda_{1}\geq|\lambda_{j}|$, $j=2,3,4$. Let $\{\omega_{1},\omega_{2},\omega_{2},\omega_{1}\}$ be a list of nonnegative numbers such that the following conditions are satisfied:

$i)$ $0\leq\omega_{k}\leq\lambda_{1}$ $k=1,2$,

$ii)$ $\omega_{1}+\omega_{2}=\frac{1}{2}(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})$,

$iii)$ $\omega_{k}\geq\lambda_{k+2}$  $k=1,2$,

$iv)$ $(2\omega_{1}-\lambda_{3})(2\omega_{2}-\lambda_{4})\geq\lambda_{1}\lambda_{2}$.

Then, there exists a centrosymmetric nonnegative matrix with espectrum $\Lambda$ and diagonal entries $\omega_{1},\omega_{2},\omega_{2},\omega_{1}$.*

Proof. Let $\Lambda$ be particioned as $\Lambda=\Lambda_{1}\cup\Lambda_{2}$, with $\Lambda_{1}=\{\lambda_{1},\lambda_{2}\}$ and $\Lambda_{2}=\{\lambda_{3},\lambda_{4}\}$.We set

[TABLE]

with spectrum $\Lambda_{1}$ and $\Lambda_{2}$ respectively. Then

[TABLE]

[TABLE]

are both nonnegative matrices and

[TABLE]

is centrosymmetric nonnegative matrix with spectrum $\Lambda$ and diagonal entries $\omega_{1},\omega_{2},\omega_{2},\omega_{1}$.

Note that in the above theorem, if $\Lambda$ is a list of real numbers of Suleimanova type, then the conditions $iii)$ and $iv)$ are always satisfied. Then $i)$ and $ii)$ become necessary and sufficient conditions.

 

In [8, Theorem 3.1] the authors show that $\lambda_{1}+\lambda_{2}-2|a|\geq 0$ and $\lambda_{1}-\lambda_{2}-2b\geq 0$ are necessary and sufficient conditions for the list $\{\lambda_{1},\lambda_{2},a+ib,a-ib\}$, with $\lambda_{1},\lambda_{2},a\in\mathbb{R},b>0$ to be the spectrum of a normal centrosymmetric nonnegative matrix. Now, we give a sufficient condition for the existence and construction of a centrosymmetric nonnegative matrix with prescribed spectrum and diagonal entries.

Theorem 3.8

*Let $\Lambda=\{\lambda_{1},\lambda_{2},a+ib,a-ib\}$ be a list of complex numbers with $\lambda_{1},\lambda_{2},a\in\mathbb{R}$, $b>0$, $\lambda_{1}+\lambda_{2}-2|a|\geq 0$ and $\lambda_{1}-\lambda_{2}-2b\geq 0$. Let $\{\omega_{1},\omega_{2},\omega_{2},\omega_{1}\}$ be a list of nonnegative numbers, such that the following conditions are satisfied:

$i)$ $0\leq\omega_{k}\leq\lambda_{1}$, $k=1,2$,

$ii)$ $\frac{1}{2}(\lambda_{1}+\lambda_{2}+2a)=\omega_{1}+\omega_{2}$,

$iii)$ $(2\omega_{1}-a)(2\omega_{2}-a)\geq\lambda_{1}\lambda_{2}+b^{2}$,

$iv)$ $\omega_{k}\geq a$, $k=1,2$.

Then, there exists a centrosymmetric nonnegative matrix with spectrum $\Lambda$ and diagonal entries $\omega_{1},\omega_{2},\omega_{2},\omega_{1}$.*

Proof. Let $\Lambda$ be particioned as $\Lambda=\Lambda_{1}\cup\Lambda_{2}$ with $\Lambda_{1}=\{\lambda_{1},\lambda_{2}\}$, $\Lambda_{2}=\{a+ib,a-ib\}$. We set

[TABLE]

with spectrum $\Lambda_{1}$ and $\Lambda_{2}$, respectively. Then

[TABLE]

[TABLE]

are both nonnegative matrices, and

[TABLE]

is nonnegative centrosymmetric matrix with spectrum $\Lambda$ and diagonal entries $\omega_{1},\omega_{2},\omega_{2},\omega_{1}$.    

Note that in the above theorem, if $\Lambda$ is a list of complex numbers of Suleimanova type, then the conditions $\lambda_{1}+\lambda_{2}-2|a|\geq 0$ and $\lambda_{1}-\lambda_{2}-2b\geq 0$ are always satisfied. Hence, for $n=4,$ a list of complex numbers of Suleimanova type is always realizable by a centrosymmetric matrix. Moreover, conditions $iii)$ and $iv)$ are always satisfied for this kind of lists. Then, conditions $i)$ and $ii)$ become necessary and sufficient conditions.

4 Examples

Example 4.1

Let $\Lambda=\{20,-1,-2,-3,-2+2i,-2-2i,-3+i,-3-i,-1+i,-1-i\}$. We shall use Corollary 3.1 to construct a centrosymmetric nonnegative matrix with spectrum $\Lambda$. Then we consider the list

[TABLE]

partitioned as $\Lambda^{\prime}=\Lambda_{1}\cup\Lambda_{2}$ with $\Lambda_{1}=\{18,-1,-2,-2+2i,-2-2i\}$, $\Lambda_{2}=\{-3,-3+i,-3-i,-1+i,-1-i\}.$ Then

[TABLE]

has spectrum $\Lambda_{2}$. Now, from Lemma 1.3, we construct the nonnegative matrix

[TABLE]

with spectrum $\Lambda_{1}$ and diagonal entries $3,3,3,1,1.$ Then

[TABLE]

and

[TABLE]

is centrosymmetric nonnegative with spectrum $\Lambda^{\prime}.$ Then

[TABLE]

is centrosymmetric nonnegative with spectrum $\Lambda$.

Example 4.2

Let $\Lambda=\{10,3,1+i,1-i,-2+2i,-2-2i,-2+2i,-2-2i\}$. Now we use Theorem 3.4 and Theorem 3.8 to construct a centrosymmetric nonnegative matrix with spectrum $\Lambda.$ We take the partition

[TABLE]

[TABLE]

and consider the associated realizable lists

[TABLE]

with realizing matrices

[TABLE]

Then

[TABLE]

From Theorem 3.8 we compute the centrosymmetric nonnegative matrix with spectrum $\Lambda_{0}$ and diagonal entries $\{4,\frac{7}{2},\frac{7}{2},4\}$

[TABLE]

Then

[TABLE]

is centrosymmetric nonnegative. Therefore by Theorem 3.4

[TABLE]