A nonpolyhedral cone of class function inequalities for positive semidefinite matrices

✍ Scribed by Wayne Barrett; H. Tracy Hall; Raphael Loewy

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 150 KB
• Volume: 302-303
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00202-5

✦ Synopsis

A function f from the symmetric group S n into R is called a class function if it is constant on each conjugacy class. Let d f be the generalized matrix function associated with f, mapping the n-by-n Hermitian matrices to R. For example, if f (σ ) = sgn(σ ), then d f (A) = det A. Let K n (K n (R)) denote the closed convex cone of those f for which d f (A) 0 for all n-by-n positive semidefinite Hermitian (real symmetric) matrices. For n = 1, 2, 3, 4 it is known that K n and K n (R) are polyhedral and there is a finite set of "test" matrices T n (T n (R)) such that f belongs to K n (K n (R)) if and only if d f (A) 0 for each A in T n (T n (R)). We show here that K 5 and K 5 (R) are not polyhedral. Thus, for n = 5 there is no finite set of "test" matrices sufficient to establish which generalized matrix functions are nonnegative on the positive semidefinite matrices.