a) Show that $X′V^{−1}X(X′V{−1}X)^{+}X′ = X′$
for any positive definite matrix V.
b) Hence show that
if $C(R′) ⊂C(X′)$, then
$R(X′V^{−1}X)^{+}R′(R(X′V^{−1}X)^{+}R′)^{+}R = R$
for any positive definite matrix V. Where C(A) represents column space of X.
My attempt:
for a :
Let V^{-1} = GG' Then LHS = $X'GG'X(X'GG'X)^{+}X'$
Assume X'G = B, Then LHS = $BB'(BB')^{+}BG^{-1}$ = $BB'B'^{+}B^{+}BG^{-1}$ = $B(B^{+}B)'B^{+}BG^{-1}$ = $BB^{+}BB^{+}BG^{-1}$ = $BG^{-1}$ = X'
For b:
I am trying to attempt in a similar fashion Taking $V^{-1}$ = GG', but I don't know how to use the information given about row space. All I can think is R can be written as AX = R, I am unable to say if R is unique or have full rank. So my attempt till now is
LHS = $R(X'GG'X)^{+}R'(R(X'GG'X)^{+}R')^{+}R$ = $R(BB')^{+}R'(R(BB')^{+}R')^{+}R$ = $(RB'^{+})(B^{+}R')(RB'^{+}B^{+}R')^{+}R$
LHS = $(RB'^{+})(B^{+}R')(B^{+}R')^{+}(RB'^{+})^{+}R$ = $(RB'^{+})(RB'^{+})^{+}R$
I am sure I am correct till this part. I want to know what insights can be derived from the column space information given? And how this is a consequence of the first problem?