Matrix Quadratic Form

Question

Say I have:

$S Q S'$, where $Q$ is positive semi definite.

Is there a quick way to see that this matrix is positive semi definite? I can see the resulting matrix being symmetric, but not immediately positive semi definite.

Thanks.

Answer 1

Let $x$ be a vector. Then $$x'SQS'x=(x'S)Q(x'S)' \geq 0$$ since $x'S$ is a vector and Q is positive semi-definite.