Proof for Semidefinite symmetric matrix product

Question

If A is symmetric positive semidefinite, show that: (a) For any matrix B,$ BAB^T $is also positive semidefinite. (b) All the diagonal elements of A are nonnegative.

Answer 1

Recall that a matrix $A$ is positive semidefinite if and only if $v^\top A v \ge 0$ for all vectors $v$.

For (a), see if $BAB^\top$ satisfies this. (Why is $v^\top B A B^\top v$ non-negative?)

For (b), consider $v^\top A v$ where $v$ is a standard unit vector.