Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the $LU$- and $QR$-factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse $M$-matrices with symmetric, irreducible, tridiagonal inverses.

Let $G$ be a graph on $n$ vertices and let ${\lambda }_1\ge {\lambda }_2\ge ...\ge {\lambda }_n$ be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues $s_k={\sum }_{i=1}^k{\lambda }_i$, for $1\le k\le n$, and show that a typical graph has $s_k\le (e\left(G\right)+k^2)/{(0.99n)}^{1/2}$, where $e\left(G\right)$ is the number of edges of $G$. We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix.

In this paper we derive new properties complementary to an $2n\times 2n$ Brualdi-Li tournament matrix $B_{2n}$. We show that $B_{2n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.

In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair $x=(\lambda ,)$ is found by solving a nonlinear equation of the form $f\left(x\right)=0$ via a contraction argument. The set-up of the method relies on the notion of $radiipolynomials$, which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.

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)...

Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_G\left(\mu \right)$, the multiplicity of a Laplacian eigenvalue $\mu$ of $G$. As a straightforward result, $m_U\left(1\right)\le n-2$. We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_G\left(1\right)$ is nondecreasing. As applications, we get the distribution of $m_U\left(1\right)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_U\left(1\right)\in \{n-5,n-3\}$, the corresponding graphs $U$ are...

We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F\left(x\right)=ux{u}^{-1}$ or $F\left(x\right)=u{x}^t{u}^{-1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...

We consider the problem of reconstructing an $n\times n$ cell matrix $D\left(\overrightarrow{x}\right)$ constructed from a vector $\overrightarrow{x}=(x_1,x_2,\cdots ,x_n)$ of positive real numbers, from a given set of spectral data. In addition, we show that the spectra of cell matrices $D\left(\overrightarrow{x}\right)$ and $D\left(\pi \left(\overrightarrow{x}\right)\right)$ are the same for every permutation $\pi \in S_n$.

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...

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 $G=(V,E)$, $V=\{v_1,v_2,...,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^{+}=D+A$ and the normalized signless Laplacian matrix as ${ℒ}^{+}=D^{-1/2}{L}^{+}{D}^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r\left(G\right)={\gamma }_2^{+}/{\gamma }_n^{+}$ and $l\left(G\right)={\gamma }_2^{+}-{\gamma }_n^{+}$, where ${\gamma }_1^{+}\ge {\gamma }_2^{+}\ge \cdots \ge {\gamma }_n^{+}\ge 0$ are eigenvalues of ${ℒ}^{+}$. We establish sharp lower and upper bounds for the normalized signless...