Minimum eigenvalue inequalities for Z -transformations on proper and symmetric cones

Z-transformations on proper and symmetric cones Z-transformations

Motivated by the similarities between the properties of Z-matrices on Rn + and Lyapunov and Stein transformations on the semidefinite cone S+, we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Final...

Some complementarity properties of Z and Lyapunov-like transformations on symmetric cones

This article deals with some complementarity properties of Z and Lyapunov-like transformations on a symmetric cone. Similar to the results proved for Lyapunov and Stein transformations on Herm(Rn×n), we show that for Lyapunov-like transformations, the P and S properties are equivalent, and that for a Ztransformation, the S-property implies the P-property of a transformation that is (quadratical...

Positive and completely positive cones and Z-transformations

A well-known result of Lyapunov on continuous linear systems asserts that a real square matrix A is positive stable if and only if for some symmetric positive definite matrix X, AX + XA is also positive definite. A recent result of Moldovan-Gowda says that a Z-matrix A is positive stable if and only if for some symmetric strictly copositive matrix X, AX + XA is also strictly copositive. In this...

Several new inequalities for the minimum eigenvalue of M-matrices

Inequalities for the minimum eigenvalue of M-matrices

Let A be a nonsingular M -matrix, and τ(A) denote its minimum eigenvalue. Shivakumar et al. [SIAM J. Matrix Anal. Appl., 17(2):298-312, 1996] presented some bounds of τ(A) when A is a weakly chained diagonally dominant M -matrix. The present paper establishes some new bounds of τ(A) for a general nonsingular M -matrix A. Numerical examples show that the results obtained are an improvement over ...