In Exercises 17–24, A is an m × n matrix with

Chapter 7, Problem 23E

(choose chapter or problem)

In Exercises 17–24, A is an \(m \times n\) matrix with a singular value decomposition \(A=U \Sigma V^{T}\), where U is an \(m \times m\) orthogonal matrix, \(\Sigma\) is an \(m \times n\) “diagonal” matrix with r positive entries and no negative entries, and V is an \(n \times n\) orthogonal matrix. Justify each answer.

Let \(U=\left[\begin{array}{lll}\mathbf{u}_{1} & \cdots & \mathbf{u}_{m}\end{array}\right]\) and \(V=\left[\begin{array}{lll}\mathbf{v}_{1} & \cdots & \mathbf{v}_{n}\end{array}\right]\), where the \(u_i\) and \(v_i\) are as in Theorem 10. Show that

\(A=\sigma_{1} \mathbf{u}_{1} \mathbf{v}_{1}^{T}+\sigma_{2} \mathbf{u}_{2} \mathbf{v}_{2}^{T}+\cdots+\sigma_{r} \mathbf{u}_{r} \mathbf{v}_{r}^{T}\)