Bidiagonal decompositions of Vandermonde-type matrices of arbitrary rank

Rank properties of a sequence of block bidiagonal Toeplitz matrices

In the present paper, we proposed a new efficient rank updating methodology for evaluating the rank (or equivalently the nullity) of a sequence of block diagonal Toeplitz matrices. The results are applied to a variation of the partial realization problem. Characteristically, this sequence of block matrices is a basis for the computation of the Weierstrass canonical form of a matrix pencil that ...

A Rank Theorem for Vandermonde Matrices

We show that certain matrices built from Vandermonde matrices are of full rank. This result plays a key role in the construction of the “limit theory of generic polynomials”.

Bidiagonal decompositions, minors and applications

Bidiagonal Decompositions of Oscillating Systems of Vectors

We establish necessary and sufficient conditions, in the language of bidiagonal decompositions, for a matrix V to be an eigenvector matrix of a totally positive matrix. Namely, this is the case if and only if V and V −T are lowerly totally positive. These conditions translate into easy positivity requirements on the parameters in the bidiagonal decompositions of V and V −T . Using these decompo...

Penrose Inverse of Bidiagonal Matrices

Introduction. For a real m×n matrix A, the Moore–Penrose inverse A+ is the unique n×m matrix that satisfies the following four properties: AAA = A , AAA = A , (A+A)T = AA , (AA+)T = AA (see [1], for example). If A is a square nonsingular matrix, then A+ = A−1. Thus, the Moore–Penrose inversion generalizes ordinary matrix inversion. The idea of matrix generalized inverse was first introduced in ...