Exercise 7

Suppose that $A=U_r\Sigma_rV_r^T$ is a reduced singular value decomposition of the $m\times n$ matrix $A$. The matrix $A^+ = V_r\Sigma_r^{-1}U_r^T$ is called the Moore-Penrose inverse of $A$.

1. Explain why $A^+$ is an $n\times m$ matrix.

2. If $A$ is an invertible, square matrix, explain why $A^+=A^{-1}$.

3. Explain why $AA^+\bvec=\bhat$, the orthogonal projection of $\bvec$ onto $\col(A)$.

4. Explain why $A^+A\xvec=\xhat$, the orthogonal projection of $\xvec$ onto $\col(A^T)$.