We design two deterministic polynomial-time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n× n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matr...

In numerical linear algebra much attention has been paid to matrices that are sparse, i.e., containing a lot of zeros. For example, to compute the eigenvalues of a general dense symmetric matrix, this matrix is first reduced to a similar tridiagonal one using an orthogonal similarity transformation. The subsequent QR-algorithm performed on this n×n tridiagonal matrix, takes the sparse structure...

Let A and B be m n matrices. A linear operator T preserves the set of matrices on which the rank is additive if rank(A + B) = rank(A) + rank(B) implies that rank(T (A) + T(B)) = rankT (A) + rankT (B). We characterize the set of all linear operators which preserve the set of pairs of n n matrices on which the rank is additive.

We present a multi-scale version of low rank matrix decomposition. Our motivation comes from imaging applications, in which image sequences are correlated locally on several scales in space and time rather than globally. We model our data matrix as a sum of matrices, where each matrix has increasing scales of locally low-rank matrices. Using this multi-scale modeling, we can capture different s...

A family of rank-metric codes over binary fields with lengths Ns = 2, s = 0, 1, . . . , is constructed. Codes of length Ns are designed recursively from codes of length Ns−1. This provides very high degree of symmetry of code matrices. In turn, it allows to decode corrupted received matrices recursively starting with small lengths. The construction allows to use many simple algorithms for decod...

The limiting spectral distribution of large sample covariance matrices is derived under dependence conditions. As applications, we obtain the limiting spectral distributions of Spearman’s rank correlation matrices, sample correlation matrices, sample covariance matrices from finite populations, and sample covariance matrices from causal AR(1) models.

A matrix is binary if each of its entries is either 0 or 1. The binary rank of a nonnegative integer matrix A is the smallest integer b such that A = BC, where B and C are binary matrices, and B has b columns. In this paper, bounds for the binary rank are given, and nonnegative integer matrices that attain the lower bound are characterized. Moreover, binary ranks of nonnegative integer matrices...

Abstract. A matrix is binary if each of its entries is either 0 or 1. The binary rank of a nonnegative integer matrix A is the smallest integer b such that A = BC, where B and C are binary matrices, and B has b columns. In this paper, bounds for the binary rank are given, and nonnegative integer matrices that attain the lower bound are characterized. Moreover, binary ranks of nonnegative intege...