rank of a residual matrix

Question

Hi I am currently studying linear regression. My question is

Is it true that

$rank(M)=n-k$ if $M = I_n- X(X'X)^{-1}X'$ where M is (n x k) matrix and $rank(M)=k$.

I can't solve it. Thanks for your help.

Answer 1

I found it by my self. Because M is symmetric and idempotent,

$rk(M) = tr(\Lambda)$ where $M = C\Lambda C'$

$tr(M) = tr(C\Lambda C') = tr(\Lambda CC')$

$tr(M) = tr(\Lambda)$

$tr(M) = tr(I_n - X(X'X)^{-1}X') = tr(I_n)-tr(X(X'X)^{-1}X') = tr(I_n) - tr(I_k) = n-k$