Copositive and Positive Quadratic Forms on Matrices

Abstract

A real symmetric quadratic form \\(f = f(Z_1,\\ldots ,Z_n)\\) in the n non-commuting indeterminates \\(Z_1,\\ldots ,Z_n\\) is said to be d-positive (respectively, d-copositive) if for all real symmetric (respectively, positive semidefinite) \\((d \\times d)\\)-matrices \\(A_1,\\ldots ,A_n\\), the matrix \\(f(A_1,\\ldots ,A_n)\\) is positive semidefinite. When \\(d=1\\), i.e., when \\(Z_i\\) can take real numbers as values, simple characterizations of real positive and copositive symmetric quadratic forms are given, for example, by Hajja (Math Inequal Appl 6:581–593, 2003). In this paper, similar characterizations are obtained for all d.