Testing Matrix inequalities

Question

I was wondering how one might check whether matrix inequalities are true/false the same way it can be done for scalar variables (like in this post).

Say I want to test the obvious inequality

A,B are positive definite dxd matrices. Prove that Tr(A B)>=0

I would do something like

conditions = {A 
\!\(\*UnderscriptBox[\(\[VectorGreater]\), 
TemplateBox[{"2"},
"SemidefiniteConeList"]]\) 0 , B 
\!\(\*UnderscriptBox[\(\[VectorGreater]\), 
TemplateBox[{"2"},
"SemidefiniteConeList"]]\) 0 , (A | B) \[Element] 
 Matrices[{d, d}, Reals, Symmetric[{1, 2}]]}

Simplify[Reduce[
  Flatten[{Tr[A.B] >= 0, conditions}], {A, B}], conditions]

But Reduce doesn't seem able to resolve matrix domains. How can such inequalities be tested?