Not all GKK $\tau$matrices are stable
Abstract
Hermitian positive definite, totally positive, and nonsingular Mmatrices 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) 12], (A), (C) implies (D) [G.M. Engel and H. Schneider, Linear and Multilinear Algebra 4 (1976) 155176], (A), (B) implies a property stronger than (D) [R. Varga, Numerical Methods in Linear Algebra, 1978, pp. 515], (A), (B), (C) implies (D) [D. Hershkowitz, Linear Algebra Appl. 171 (1992) 161186]. We describe a class of unstable GKK $\tau$matrices, thus disproving all four conjectures.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2005
 arXiv:
 arXiv:math/0512586
 Bibcode:
 2005math.....12586H
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Numerical Analysis;
 15A18;
 15A42;
 34D20;
 93D05;
 47A75;
 47B35
 EPrint:
 7 pages