Let $ M $ be an $ n\times n $ Hermitian matrix of rank $ k $, $k\neq n$. If $\lambda $ $\neq 0$ is an eigenvalue of $ M $ with corresponding unit column vector $ u $, with $ Mu= \lambda u$,then Which of the following are true?
rank ($ M-\lambda $ $uu^* $) =$ k-1 $
$ ( M- \lambda uu^* )^n = M^n- \lambda^n uu^* $
I think both are true, I considered some examples which is obviously not the right way to solve the above problem. I tried to use the fact that any Hermitian matrix is unitarily diagonalizable to show the 2nd option , but I couldn't do it.
please give me some hints . Thanks in advance.