We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$. The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$. Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show that the choice where $X$ is the all-ones matrix allows to estimate the signature of the leading eigenvector of $A$, generalizing previous results on Perron-Frobenius properties of matrices...

A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_1$ and $D_2$ such that $A^{-\mathrm{T}}=D_{1}A{D}_2$, where $A^{-\mathrm{T}}$ denotes the transpose of the inverse of $A$. Denote by $J=\mathrm{diag}(\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+1$ or $-1$. A nonsingular real matrix $Q$ is called $J$-orthogonal if $Q^{\mathrm{T}}JQ=J$. Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

Let $A$, $B$ be $n\times m$ matrices. The concept of matrix majorization means the $j$th column of $A$ is majorized by the $j$th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$. We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

Consider the $n\times n$ matrix with $(i,j)$’th entry $gcd(i,j)$. Its largest eigenvalue ${\lambda }_n$ and sum of entries $s_n$ satisfy ${\lambda }_n>s_n/n$. Because $s_n$ cannot be expressed algebraically as a function of $n$, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that ${\lambda }_n>6{\pi }^{-2}nlogn$ for all $n$. If $n$ is large enough, this follows from F. Balatoni (1969).

Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^{4}lt;\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the...

By max-plus algebra we mean the set of reals $ℝ$ equipped with the operations $a\oplus b=max\{a,b\}$ and $a\otimes b=a+b$ for $a,b\in ℝ.$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A,B\in ℝ(m,n)$ if $A\otimes x=\lambda \otimes B\otimes x$ for some $\lambda \in ℝ$. The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval)...

Consider a $N\times n$ non-centered matrix ${𝛴}_n$ with a separable variance profile: $${𝛴}_n=\frac{D_n^{1/2}{X}_n{\tilde{D}}_n^{1/2}}{\sqrt{n}}+A_n.$$ Matrices $D_n$ and ${\tilde{D}}_n$ are non-negative deterministic diagonal, while matrix $A_n$ is deterministic, and $X_n$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_n\left(z\right)$ the resolvent associated to ${𝛴}_n{𝛴}_n^{*}$, i.e. $$Q_n\left(z\right)={\left({𝛴}_n{𝛴}_n^{*}-z{I}_N\right)}^{-1}.$$ Given two sequences of deterministic vectors $\left(u_n\right)$ and $\left(v_n\right)$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:...

A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A\otimes x=x$. An eigenvector $x$ of $A$ is called the greatest $𝐗$-eigenvector of $A$ if $x\in 𝐗=\{x;\underline{x}\le x\le \overline{x}\}$ and $y\le x$ for each eigenvector $y\in 𝐗$. A max-min matrix $A$ is called strongly $𝐗$-robust if the orbit $x,A\otimes x,A^2\otimes x,\cdots$ reaches the greatest $𝐗$-eigenvector with any starting vector of $𝐗$. We suggest an $O\left(n^3\right)$ algorithm for computing the greatest $𝐗$-eigenvector of $A$ and study the strong $𝐗$-robustness. The necessary and sufficient conditions for strong $𝐗$-robustness are introduced...

Let $G$ be a simple connected graph with vertex set $V\left(G\right)=\{v_1,v_2,\cdots ,v_n\}$ and edge set $E\left(G\right)$, and let $d_{v_i}$ be the degree of the vertex $v_i$. Let $D\left(G\right)$ be the distance matrix and let $T_r\left(G\right)$ be the diagonal matrix of the vertex transmissions of $G$. The generalized distance matrix of $G$ is defined as $D_{\alpha }\left(G\right)=\alpha T_r\left(G\right)+(1-\alpha )D\left(G\right)$, where $0\le \alpha \le 1$. Let ${\lambda }_1\left(D_{\alpha }\left(G\right)\right)\ge {\lambda }_2\left(D_{\alpha }\left(G\right)\right)\ge ...\ge {\lambda }_n\left(D_{\alpha }\left(G\right)\right)$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\le k\le n$. We denote by $S_k\left(D_{\alpha }\left(G\right)\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)+{\lambda }_2\left(D_{\alpha }\left(G\right)\right)+...+{\lambda }_k\left(D_{\alpha }\left(G\right)\right)$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{\alpha }S\left(G\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)-{\lambda }_n\left(D_{\alpha }\left(G\right)\right)$....

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity $\Phi ={\sum }_{i+k=j+l}\frac{q_i{q}_k}{q_j{q}_l}$ satisfies $\Phi \ge N^2$, with equality if and only if $q=\left(q_i\right)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi$, (2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We explore here...

Let $m$ and $n$ be positive integers, and let $R=(r_1,...,r_m)$ and $S=(s_1,...,s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m\times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F\left(R\right)$ be the $m\times n$ $(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, $\mathrm{disc}\left(A\right)$, is the number of positions in which $F\left(R\right)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over...

An $m\times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let ${𝐌}_{m,n}$ be the set of all $m\times n$ real matrices. For $A,B\in {𝐌}_{m,n}$, we say that $A$ is row Hadamard majorized by $B$ (denoted by $A{\prec }_{RH}B)$ if there exists an $m\times n$ row stochastic matrix $R$ such that $A=R\circ B$, where $X\circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in {𝐌}_{m,n}$. In this paper, we consider the concept of row Hadamard majorization as a relation on ${𝐌}_{m,n}$ and characterize the structure of all linear operators $T:{𝐌}_{m,n}\to {𝐌}_{m,n}$ preserving...