Not all GKK τ-matrices are stable

Hermitian positive definite, totally positive, and nonsingular M-matrices enjoy many common properties, in particular: (A) positivity of all principal miners, (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) double right arrow (D) [D. Carlson, J. Res. Nat. Bur. Standards Sect. B 78 (1974) 1-2], (A), (C) double right arrow (D) [G.M. Engel and H. Schneider, Linear and Multilinear Algebra 4 (1976) 155-176], (A), (B) double right arrow a property stronger than (D) [R. Varga, Numerical Methods in Linear Algebra, 1978, pp. 5-15], (A), (B), (C) double right arrow (D) [D. Hershkowitz, Linear Algebra Appl. 171 (1992) 161-186]. We describe a class of unstable GKK tau-matrices, thus disproving all four conjectures. (C) 1999 Elsevier Science Inc. All rights reserved.