(Moore-Penrose pseudo-inverse)
Let \(A \in \mathbb {C}^{m \times n}\) have rank \(r\) and reduced SVD \[ A = \tilde {U}\tilde {\Sigma }\tilde {V}^* . \]
The Moore-Penrose pseudoinverse of \(A\) is defined as
\[ A^+ = \tilde {V}\tilde {\Sigma }^{-1}\tilde {U}^* . \]
a. In the induced matrix \(\ell ^2\) norm \(\|\cdot \|\), is the operation \(A \mapsto A^+\) well-conditioned? That is, given an arbitrary but fixed \(A\) and a perturbation matrix \(B\), is \(\|(A + B)^+ - A^+\|/\|A^+\|\) controllable by \(\|B\|/\|A\|\)? Prove it, or give a counterexample.
b. If \(m \ge n\) and \(A\) has full rank \(n\) and \(b \in \mathbb {C}^m\) is a given vector, prove that \(x = A^+b\) is identical to the least-squares solution \(z\) of the over-determined linear system \(Az = b\).