  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.
