The Positive Minorant Property on Matrices

We study the positive minorant property for norms on spaces of matrices. A matrix is said to be a majorant of another if all the entries in the first matrix are greater than or equal to the absolute values of the corresponding entries in the second matrix. For a real number p ≥ 0 the Schatten p-norm of the matrix is the l^p-norm of its singular values. The space of n X n matrices with the Schatten p-norm is said to have the positive minorant property if the norm of each nonnegative matrix is greater than or equal to the norm of every nonnegative matrix that it majorizes. It is easy to show that this property holds if p is even. We show that the positive minorant property fails when p < 2(n - 1) and p not even, and provide a simple proof to show the property does hold when p ≥ 2(n - 1)[(n - 1)/2] + 2.