Let A ∈ℂ^p ×p, B ∈ℂ^p ×q, M=[
A B
B^H O
] and suppose that B B^H=Ip. Show that

Question

Let A ∈ℂ^p ×p, B ∈ℂ^p ×q, M=[
    A     B 
     B^H     O
] and suppose that B B^H=Ip. Show that

Accepted Answer

Step 1: Recall the definition of the Moore-Penrose inverse.
The Moore-Penrose inverse $M^dagger$

Let $A \in \mathbb{C}^{p \times p}, B \in \mathbb{C}^{p \times q}$, $$ M=\left[\begin{array}{cc} A & B \\ B^H & O \end{array}\right] $$ and suppose that $B B^H=I_p$. Show that the Moore-Penrose inverse $$ M^{\dagger}=\left[\begin{array}{cc} O & B \\ B^H & -B^H A B \end{array}\right] . $$