Math 6610 • Analysis of Numerical Methods I

(Properties via SVD). Suppose \(A \in \mathbb {C}^{m \times n}\) has an SVD, \(A = U\Sigma V^*\), where \(u_j\) are the left singular vectors, \(U = \{u_j\}\), \(v_j\) are the right singular vectors, \(V = \{v_j\}\), and \(\sigma _j\) are the singular values. Assume rank of the matrix \(A\) is \(r\). For any \(\nu \) with \(0 \le \nu \le r\), define \[ A_\nu = \sum _{j=1}^{\nu }\sigma _j u_j v_j^*, \] (if \(\nu = p = \min \{m,n\}\), define \(\sigma _{\nu +1} = 0\)). Show that, \[ \|A - A_\nu \|_2 = \inf _{\substack {B \in \mathbb {C}^{m \times n}\\ \operatorname {rank}(B) \le \nu }} \|A - B\|_2 = \sigma _{\nu +1}. \]