Let $X$ be an $(n \times n)$ matrix. Let $V$ be the $(n \times n-k)$ be the matrix of eigenvectors of $X$ which correspond to non-zero eigenvalues of $X$. Let $E$ be the $(n-k \times n-k)$ diagonal matrix of eigenvalues of $X$. I wish to prove that:
I can prove it when there are no zero-valued eigenvalues but not the general case. Would anyone be able to help?