Proving a theorem about trace of matrix which involving generalized inverse matrix

Question

can you prove that theorem for me:

Let A be mxn matrix of rank r then,

$\ tr[I-A(A'A)^-A'] = m-r $  .  

$\ A' $(transpose of A) ,$\ A^- $(generalized inverse of A)

Answer 1

Because $A'A$ also has rank $r$, there are matrices $B$ ($n\times r$) and $C$ ($r\times n$) of rank $r$ satisfying $A'A=B\times C$. Then, $$ (A'A)^-=C'(CC')^{-1}(B'B)^{-1}B'. $$ It follows that $$ \text{Tr}(A(A'A)^-A')=\text{Tr}(A'A(A'A)^-)=\text{Tr}[BCC'(CC')^{-1}(B'B)^{-1}B']\\ =\text{Tr}[B(B'B)^{-1}B']=\text{Tr}[B'B(B'B)^{-1}]=\text{Tr}(I_r)=r. $$ The conclusion now follows: $$ \text{Tr}(I_m-A(A'A)^-A')=\text{Tr}(I_m)-\text{Tr}(A(A'A)^-A')=m-r. $$

Answer 2

@Kim Jong Un

$\ (A'A)^-=C'(CC')^{-1}(B'B)^{-1}B'$ Can you explain this is coming from where? Is it a property ? Can you give a link about this ? Thanks