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

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 P is an orthogonal m x m matrix, then PA has the same singular values as A.