Chapter 7.4, Problem 22E

In Exercises 17–24, A is an m × n matrix with a singular value decomposition A = UΣV^T, where U is an m × m orthogonal matrix, Σ is an m × n “diagonal” matrix with r positive entries and no negative entries, and V is an n × n orthogonal matrix. Justify each answer.

22. Show that if A is an n × n positive definite matrix, then an orthogonal diagonalization A = PDP^T is a singular value decomposition of A.