Related to the trace of the product of two symmteric matrices

Question

I have two symmetric matrices $A$ and $B$. One of them ($A$) is positive semidefinite and I want $$Tr(AB)\geq 0.$$ In this case how to show that $B$ must be a positive semidefinite matrix. Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

1) In $\mathbb{R}^n$, $e\neq 0$ is a vector. If $X:=\{v\in \mathbb{R}^n| v\cdot e\geq 0\}$, then for any $x\in X$, $$ x-ce\in e^\perp$$ for some $c\geq 0$.

2) Note that $M_n(\mathbb{R})$ is a vector space. Here we can give an inner product $i$ s.t. $i(A,B)={\rm Tr}\ (AB^T)$ where $T$ is a transpose. Hence remaining is similar to the previous case