2.2 Traces of products
Let \(\iota \) be a finite set; all matrices below are in \(\mathbb {C}^{\iota \times \iota }\).
(a) For positive semidefinite \(P\) and \(Q\) we have \(\operatorname {Tr}(P Q) \geq 0\). If \(P\) is positive definite, \(Q\) is positive semidefinite and \(Q \neq 0\), then \(\operatorname {Tr}(P Q) {\gt} 0\).
(b) Let \(Y\) be positive semidefinite and \(Z\) any matrix. Then
(a) Write \(Q\) as a sum of rank-one matrices \(v v^*\). Then \(\operatorname {Tr}(P v v^*) = v^* P v \geq 0\), with strict inequality for \(v \neq 0\) when \(P\) is positive definite, and at least one of the \(v\) is nonzero when \(Q \neq 0\).
(b) A positive semidefinite \(Y\) makes \(\langle P, Q \rangle = \operatorname {Tr}(Q Y P^*)\) a semi-inner product on matrices. Cauchy–Schwarz for the pair \(1\), \(Z\) is the inequality. No commutation of \(Y\) and \(Z\) is used.