This question was asked in my masters exam for which I am preparing and I was unable to solve it.
Let M be a $n\times n$ Hermitian matrix of rank $k (k<n)$ . If $\lambda \neq 0$ is an eigenvalue of $M$ with corresponding unit column vector u then which of the following are true.
A. Rank$(M-\lambda u u*)$=k-1.
B. Rank$(M-\lambda u u*)$=k.
C. Rank$(M-\lambda u u*)$=k+1.
D. Rank${(M-\lambda u u*)}^n= M^n- {\lambda}^n uu*$.
I am sorry I would not be able to give anything as attempt as I am not really good at problem solving in linear algebra. Also, I could not decide which result I should use. I am blank on this question.
Looking for your help.