Basic matrices

Complementary basic matrices

We show that an n× n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions (except the first and last) completely between the subdiagonal and superdiagonal part, then this matrix can be factorized into a product of n− 1 matrices, each consisting of a 2 × 2 principal submatrix in two consecutive rows (and columns) in all possible of the...

Basic Properties of Circulant Matrices and Anti-Circular Matrices

For simplicity, we adopt the following convention: i, j, k, n, l denote elements of N, K denotes a field, a, b, c denote elements of K, p, q denote finite sequences of elements of K, and M1, M2, M3 denote square matrices over K of dimension n. Next we state two propositions: (1) 1K · p = p. (2) (−1K) · p = −p. Let K be a set, let M be a matrix over K, and let p be a finite sequence. We say that...

Linear Discrepancy of Basic Totally Unimodular Matrices

We show that the linear discrepancy of a basic totally unimodular matrix A ∈ Rm×n is at most 1− 1 n+1 . This extends a result of Peng and Yan. AMS Subject Classification: Primary 11K38.

Basic Properties of the Rank of Matrices over a Field

In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field. I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m× n matrix A by th...

Basic Relationships for Matrices Describing Scattering by Small Particles