Not all GKK τ-matrices are stable

Abstract

Hermitian positive definite, totally positive, and nonsingular M-matrices enjoy many common properties, in particular:

• positivity of all principal minors,

• weak sign symmetry,

• eigenvalue monolonicity,

• positive stability.

The class of GKK matrices is defined by properties (A) and (B), whereas the class of nonsingular τ-matrices by (A) and (C). It was conjectured that:

• (A), (B) ⇒ (D) [D. Carlson, J. Res. Nat. Bur. Standards Sect. B 78 (1974) 1–2],

• (A), (C) ⇒ (D) [G.M. Engel and H. Schneider, Linear and Multilinear Algebra 4 (1976) 155–176].

• (A), (B) ⇒ a property stronger than (D) [R. Varga, Numerical Methods in Linear Algebra, 1978, pp. 5–15],

• (A), (B), (C) ⇒ (D) [D. Hershkowitz, Linear Algebra Appl. 171 (1992) 161–186].

We describe a class of unstable GKK τ-matrices, thus disproving all four conjectures.