  We investigate estimation of a normal mean matrix under the matrix quadratic
loss. Improved estimation under the matrix quadratic loss implies improved
estimation of any linear combination of the columns. First, an unbiased
estimate of risk is derived and the Efron--Morris estimator is shown to be
minimax. Next, a notion of \textit{matrix superharmonicity} for matrix-variate
functions is introduced and shown to have analogous properties with usual
superharmonic functions, which may be of independent interest. Then, we show
that the generalized Bayes estimator with respect to a matrix superharmonic
prior is minimax. We also provide a class of matrix superharmonic priors that
includes the previously proposed generalization of Stein's prior. Numerical
results demonstrate that matrix superharmonic priors work well for low rank
matrices.
