In Exercises 17–24, A is an m × n matrix with
Chapter 7, Problem 21E(choose chapter or problem)
In Exercises 17-24, A is an m X n matrix with a singular value decomposition \(A=U \Sigma V^T\), where U is an m X m orthogonal matrix, \(\Sigma\) is an m X n "diagonal" matrix with r positive entries and no negative entries, and V is an n X n orthogonal matrix. Justify each answer.
Justify the statement in Example 2 that the second singular value of a matrix A is the maximum of \(\|A \mathbf{x}\|\) as x varies over all unit vectors orthogonal to \(v_1\), with \(v_1\) a right singular vector corresponding to the first singular value of A. [Hint: Use Theorem 7 in Section 7.3.]
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