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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.

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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.

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