A is an m x n matrix with a singular value decomposition A = UV T ,

A is an m x n matrix with a singular value decomposition A = UΣV^T, where U is an m x m orthogonal matrix, Σ 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.

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