LOEWNER MATRIX ORDERING IN ESTIMATION OF THE SMALLEST SINGULAR VALUE ∗

Abstract. In this paper, some new lower bounds for the smallest singular value of a square complex matrix A are derived. A key tool to obtain these bounds is using some Hermitian matrices which are, in the sense of Loewner matrix ordering, below the Hermitian part of A or, more generally, below the Hermitian part of the unitary equivalences of A. Two types of new bounds are proposed. The first bound can be applied to matrices with positive diagonal entries and strictly diagonally dominant Hermitian parts. It is always at least as large (never worse) as pure Gersgorin-based bound due to C.R. Johnson [C.R. Johnson. A Gersgorin-type lower bound for the smallest singular value. Linear Algebra Appl., 112:1–7, 1989.]. The other bound is complementary with the first one and it can be effectively applied to matrices whose Hermitian parts are very far from diagonal dominance

Bidiagonal decompositions, minors and applications

Abstract. Matrices, called ε-BDmatrices, that have a bidiagonal decomposition satisfying some sign constraints are analyzed. The ε-BD matrices include all nonsingular totally positive matrices, as well as their matrices opposite in sign and their inverses. The signs of minors of ε-BD matrices are analyzed. The zero patterns of ε-BD matrices and their triangular factors are studied and applied to prove the backward stability of Gaussian elimination without pivoting for the associated linear systems. Key words. Bidiagonal decomposition, totally positive matrices, Gaussian elimination, backward stability. AMS subject classifications. 15A18, 15A15, 65F05, 65F15

CHARACTERIZATIONS AND DECOMPOSITIONS OF ALMOST STRICTLY POSITIVE MATRICES ∗

Abstract. A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative, and furthermore these minors are positive if and only if their diagonal entries are positive. In this paper we give a characterization of these matrices in terms of the positivity of a very reduced number of their minors (which are called boundary minors), improving previous characterizations that have appeared in the literature. We show the role of boundary minors in accurate computations with almost strictly totally positive matrices. Moreover, we analyze the QR factorization of these matrices, showing the differences and analogies with that of totally positive matrices. Key words. total positivity, QR factorization, almost strictly totally positive matrices AMS subject classifications. 65F25, 65F40, 15A4