LAUSR

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.