Related papers: Inertia of Loewner Matrices

Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $$\rho, b+ic,b-ic, \lambda_4,\cdots,\lambda_n,$$ where $\rho$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a…

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of this space where the maximal dimension is attained. Extensions of this result in the direction of linear…

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the…

Let ${\bf A} \in R^{n \times n}$ be a nonnegative irreducible square matrix and let $r({\bf A})$ be its spectral radius and Perron-Frobenius eigenvalue. Levinger asserted and several have proven that $r(t):=r((1{-}t) {\bf A} + t {\bf…

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…

An $n\times n$ matrix is said to have a self-interlacing spectrum if its eigenvalues $\lambda_k$, $k=1,\ldots,n$, are distributed as follows $$ \lambda_1>-\lambda_2>\lambda_3>\cdots>(-1)^{n-1}\lambda_n>0. $$ A method for constructing sign…

In a 1997 paper Lin solved a longstanding problem as follows: For each epsilon > 0, there is delta > 0 such that if h and k are self-adjoint contractive n x n matrices and ||hk - kh|| < delta, then there are commuting self-adjoint matrices…

Given a positive integer $r$ and a prime power $q$, we estimate the probability that the characteristic polynomial $f_{A}(t)$ of a random matrix $A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic) irreducible factors…

A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with…

For any primitive matrix $M\in\mathbb{R}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots,x_n)^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots,\frac{1}{x_n})^t$.…

Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\…

We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and…

For positive semidefinite $n\times n$ matrices $A$ and $B$, the singular value inequality $(2+t)s_{j}(A^{r}B^{2-r}+A^{2-r}B^{r})\leq 2s_{j}(A^{2}+tAB+B^{2})$ is shown to hold for $r=\frac{1}{2}, 1, \frac{3}{2}$ and all $-2<t\leq 2$.

For a real $N\ge 1$ and a vector $\xi =(1,\xi_1,...,\xi_n)$ define a matrix $$ {\cal A} (\xi, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \cr N^{\frac{1}{n}} \xi_1 & -N^{\frac{1}{n}} & 0&... & 0 \cr N^{\frac{1}{n}} \xi_2 &0& -N^{\frac{1}{n}}…

The main purpose of this paper is to show that there exists a positive number $\lambda_{1}$, the first eigenvalue, such that some $p(x)$-Laplace equation admits a solution if $\lambda=\lambda_{1}$ and that $\lambda_{1}$ is simple, i.e.,…

Let H be any complex inner product space with inner product <, >. We say that f : C -->C is Hermitian positive definite on H if the matrix $$(f(<z^r,z^s>))_{r,s=1}^n \eqno(*)$$ is Hermitian positive definite for all choice of z^1,...,z^n in…

We present an explicit polynomial formula for evaluating the principal logarithm of all matrices lying on the line segment $\{I(1-t)+At:t\in [0,1]\}$ joining the identity matrix $I$ (at $t=0$) to any real matrix $A$ (at $t=1$) having no…

We give a complete characterization of nonnegative integers $j$ and $k$ and a positive integer $n$ for which there is an $n$-by-$n$ matrix with its power partial isometry index equal to $j$ and its ascent equal to $k$. Recall that the power…

It is well known that the following Collatz Conjecture is one of the unsolved problems in mathematics. Collatz Conjecture: For any positive integer $n>1$, the following recursive algorithm will convergent to 1 by a finite number of steps.…

A. Albouy and R. Moeckel in 2000 found some interesting inequalities related to the inverse problem for collinear (Moulton) central configurations: the Pfaffian of a certain matrix is positive since all coefficients of some polynomials are…