Not all GKK $\tau$-matrices are stable

Abstract

Hermitian positive definite, totally positive, and nonsingular M-matrices enjoy many common properties, in particular: (A) positivity of all principal minors, (B) weak sign symmetry, (C) eigenvalue monotonicity, (D) positive stability. The class of GKK matrices is defined by properties (A) and (B), whereas the class of nonsingular $\tau$-matrices by (A) and (C). It was conjectured that: (A), (B) implies (D) [D. Carlson, J. Res. Nat. Bur. Standards Sect. B 78 (1974) 1-2], (A), (C) implies (D) [G.M. Engel and H. Schneider, Linear and Multilinear Algebra 4 (1976) 155-176], (A), (B) implies a property stronger than (D) [R. Varga, Numerical Methods in Linear Algebra, 1978, pp. 5-15], (A), (B), (C) implies (D) [D. Hershkowitz, Linear Algebra Appl. 171 (1992) 161-186]. We describe a class of unstable GKK $\tau$-matrices, thus disproving all four conjectures.

7 pages

Cited by in corpus (4)

• Hyperdeterminantal relations among symmetric principal minors
• Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices
• Open problems on GKK tau-matrices
• Theorems and counterexamples on structured matrices