Matrix quadratic risk of orthogonally invariant estimators for a normal mean matrix

In estimation of a normal mean matrix under the matrix quadratic loss, we develop a general formula for the matrix quadratic risk of orthogonally invariant estimators. The derivation is based on several formulas for matrix derivatives of orthogonally invariant functions of matrices. As an application, we calculate the matrix quadratic risk of a singular value shrinkage estimator motivated by Stein's proposal for improving on the Efron--Morris estimator 50 years ago.