Bidiagonal decompositions, minors and applications

Abstract

Abstract. Matrices, called ε-BD matrices, 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.

Bidiagonal decompositions and total positivity of some special matrices

“…Careful analyses of the relationships between totally nonnegative matrices and bidiagonal decompositions have been done in [10,12]. For more results on bidiagonal decompositions of matrices, see [1,15,16,19]. In particular, in the case of invertible totally nonnegative matrices, the unicity of the bidiagonal decomposition under certain conditions was assured in [12].…”

All Jordan canonical forms of irreducible totally non-negative matrices

“…A ∈ R n×n is called totally nonnegative if all its minors are nonnegative and it is abbreviated as TN, see for instance [1][2][3][4][5]. Due to its wide variety of applications in algebra, computer aided geometric design, differential equations, economics, quantum theory and other fields, TN matrices have been studied by several authors who have obtained properties, the Jordan structure and characterizations by using the Neville elimination [5][6][7][8][9][10].…”

“…A(w + i, 1 : 11) = [A(j, 8) + i, A(j, 2) − 1, A(j, 1) − (A(j, 8) + i), (A(j, 2) − 1) * (A(j, 1) − (A(j, 8) + i)), zeros (1,6) A(q, 10) = A(q, 10) * (−1) A(q,11) ; 47: end for 48: c =ones(1, j) * A(1 : j, 10);…”