Using SVD to prove that a matrix X will be all zeros if X'X is a matrix with all zeros

Question

I have a question like this: We need to use the SVD to prove that if X'X = 0(a matrix with all zeros), where ' means a transpose, then the matrix X = 0; I think may be I need to prove the singular value matrix Sigma will be a matrix with all zeros. Thanks!

Answer 1

$X = U\Sigma V'$.

$0=X'X=V\Sigma^2 V$.

Multiply on the right by $V'$ and on the left by $V'$ to conclude $\Sigma^2=0$, which implies $\Sigma=0$ since it is non-negative.