Copositive and positive quadratic forms on matrices (Q2274924)

The authors call a quadratic form \(f\), in \(n\) noncommuting variables \(Z_1, \dots , Z_n\), \(d\)-positive if \(f (A_1 , \dots, A_n ) \geq0\) for all symmetric \(d \times d\) matrices, and \(d\)-copositive if \(f(A_1,\dots A_n)\geq 0\) for all positive definite \(d\times d\) matrices. They define \(T = \sum_{1\leq i \leq n} Z_i^2\) and \(W = \sum_{1\leq {i,j} \leq n, \, i\neq j}Z_i Z_j\) and write \(f = xT + yW\). The main result that the authors obtain are necessary and sufficient conditions for \(F\) to be positive and also for \(F\) to be copositive.