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.

翻译：在矩阵二次损失下估计正态均值矩阵时，我们推导了正交不变估计量矩阵二次风险的一般公式。该推导基于几个关于矩阵正交不变函数矩阵导数的公式。作为应用，我们计算了由50年前Stein提出的改进Efron-Morris估计量的方案所启发的奇异值收缩估计量的矩阵二次风险。