I saw a version of HW4 Q5, where the Moore-Penrose (MP) inverse is used throughout. I (wrongly) believed that the MP inverse can be replaced by any generalized inverse. It seems to be right, except for part (e). No one can show that the lower right block of the matrix $ZZ^-$ is $B^-B$. Trivially we know that this block should be equal to $\Lambda\Lambda^-$. But is this true that \(B^- B = \Lambda \Lambda^-?\)

The answer is negative. Shihui provided a concrete counter-example, showing that \(B^+ B = \Lambda \Lambda^+,\) but in general \(B^- B \ne \Lambda \Lambda^-.\)

Thanks, Shihui!