A note on maximizing the permanent of a positive definite hermitian matrix, given the eigenvalues
Robert D. Grone, Charles R. Johnson, SA Eduardo, Henry Wolkowicz · Linear and Multilinear Algebra · 1986
As a step toward understanding the unsolved problem of determining how large the permanent of a positive semi-definite matrix can be, given the eigenvalues, we note that a necessary condition for A to be a permanent maximizing matrix is that A commute with its permanental adjoint.