Sw. Drury, A BOUND FOR THE DETERMINANT OF CERTAIN HADAMARD-PRODUCTS AND FOR THE DETERMINANT OF THE SUM OF 2 NORMAL MATRICES, Linear algebra and its applications, 199, 1994, pp. 329-338
We establish an upper bound for the absolute value of the determinant
of the Hadamard (elementwise) product of a unitary matrix and a genera
l complex matrix. In the case that the general matrix is fixed and the
unitary matrix is allowed to vary, this estimate is best possible. As
a corollary, we obtain an upper bound for the absolute value of the d
eterminant of the sum of two normal matrices with specified eigenvalue
s. This corollary supports the determinantal conjecture of Marcus and
de Oliveira.