Related papers: The convergence of power matrices

How many random entries of an n by m, rank r matrix are necessary to reconstruct the matrix within an accuracy d? We address this question in the case of a random matrix with bounded rank, whereby the observed entries are chosen uniformly…

Given an ordered triple of positive integers $(n,r,b)$, where $1\leq b\leq\binom{n}{r}$, does there exist a matrix of size $r\times n$ with exactly $b$ invertible submatrices of size $r\times r$? Such a matrix is called an $(n,r,b)$-matrix.…

We prove a probabilistic generalization of the classic result that infinite power towers, $c^{c^{\dots}}$, converge if and only if $c\in[e^{-e},e^{1/e}]$. Given an i.i.d. sequence $\{A_i\}_{i\in\mathbb N}$, we find that convergence of the…

The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn…

Positivity properties of the Hadamard powers of the matrix $\begin{bmatrix}1+x_ix_j\end{bmatrix}$ for distinct positive real numbers $x_1,\ldots,x_n$ and the matrix $\begin{bmatrix}|\cos((i-j)\pi/n)|\end{bmatrix}$ are studied. In…

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in $l^p$ when $p \geq 1$ and its dual version, the upper bounds when $0<p…

The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy…

The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, often called the \emph{Sinkhorn limit} of $A$. The main result in this paper…

A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a…

In this short note, we study the behaviour of a product of matrices with a simultaneous renormalization. Namely, for any sequence $(A\_n)\_{n\in \mathbb{N}}$ of $d\times d$ complex matrices whose mean $A$ exists and whose norms' means are…

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…

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the…

In this article, we consider birational equivalence of exponential matrices. In characteristic zero, we give a birational classification of exponential matrices of size $n$-by-$n$ $(n \geq 2)$, which consists of two types. And in positive…

Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite and positive definite matrices of order $n$, respectively, with nonnegative entries, where some positions of zero entries are restricted…

We consider convergence sets of formal power series of the form $f(z,t)=\sum_{n=0}^{\infty} f_n(z)t^n$, where $f_n(z)$ are holomorphic functions on a domain $\Omega$ in $\mathbb{C}$. A subset $E$ of $\Omega$ is said to be a convergence set…

Necessary and sufficient conditions are given on matrices $A$, $B$ and $S$, having entries in some field $\mathbb F$ and suitable dimensions, such that the linear span of the terms $A^iSB^j$ over $\mathbb F$ is equal to the whole matrix…

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that $S(k,\ell) = \mbox{tr}(A_k^*…

Nonlinear matrix equations are encountered in many applications of control and engineering problems. In this work, we establish a complete study for a class of nonlinear matrix equations. With the aid of Sherman Morrison Woodbury formula,…

We consider a non-relativistic quantum particle in $\mathbb{R}^d$, $d=2$ or $d = 3$, interacting with singular zero-range potentials concentrated on a large collection of points. We analyze the homogenization regime where the intensities of…

The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank…