Cauchy Schwarz inequality for trace

Question

I found the following theorem in wikipedia. If matrices $A$ and $B$ are positive semi-definite of the same size then $trace(AB) \leq trace(A) trace(B)$.

I need a reference book having this theorem to which I can refer. I just want to use it. I appreciate if anybody can introduce a reference for this to me.

Answer 1

This inequality is proven, amongst numerous others, in

Z.P. Yang, X.X. Feng, "A note on the trace inequality for products of Hermitian matrix power." J. of Inequalities in Pure and App. Math. Volume 3, Issue 5, No. 78, 2002.

In particular, it is Lemma 2.3 in the article.