Related papers: A lower bound for periods of matrices

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes,…

A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer…

An n \times n matrix A has a normal defect of k if there exists an (n+k) \times (n+k) normal matrix A_{ext} with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 \times 4…

This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees…

We show that if $A$ is an $n\times n$-matrix, then the diagonal entries of each power $A^{m}$ are uniquely determined by the principal minors of $A$, and can be written as universal (integral) polynomials in the latter. Furthermore, if the…

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition…

Let K be a (commutative) field with characteristic not 2, and V be a linear subspace of n by n matrices that have at most two eigenvalues in K (respectively, at most one non-zero eigenvalue in K). We prove that the dimension of V is less…

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We…

We prove the following results: let x,y be (n,n) complex matrices such that x,y,xy have no eigenvalue in ]-infinity,0] and log(xy)=log(x)+log(y). If n=2, or if n>2 and x,y are simultaneously triangularizable, then x,y commute. In both cases…

We consider real orthogonal $n\times n$ matrices whose diagonal entries are zero and off-diagonal entries nonzero, which we refer to as $\mathrm{OMZD}(n)$. We show that there exists an $\mathrm{OMZD}(n)$ if and only if $n\neq 1,\ 3$, and…

A unimodular $2\times 2$ matrix $A$ with entries in a commutative ring $R$ is called weakly determinant liftable if there exists a matrix $B$ congruent to $A$ modulo $R\det(A)$ and $\det(B)=0$; if we can choose $B$ to be unimodular, then…

We consider n by n real matrices whose entries are non-degenerate random variables that are independent but non necessarily identically distributed, and show that the probability that such a matrix is singular is O(1/sqrt{n}). The purpose…

In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover,…

We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds…

A notable difference between the ordinary and Hadamard products is that the Hadamard product of two singular positive semidefinite matrices can be nonsingular, and one of the factors can even be indefinite. We present an eigenvalue lower…

We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$…

Let $T$ be an $n\times n$ random matrix, such that each diagonal entry $T_{i,i}$ is a continuous random variable, independent from all the other entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$ $$…

If $A$ is an integer valued, strictly expansive matrix, then there exists an orthonormal $A$-wavelet whose Fourier transform is compactly supported and smooth. We show that strongly connected diagonally dominant integer matrices are…

A binary matrix has the consecutive ones property (C1P) if it is possible to order the columns so that all 1s are consecutive in every row. In [McConnell, SODA 2004 768-777] the notion of incompatibility graph of a binary matrix was…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…