The determinant of a complex matrix and Gershgorin circles

Abstract

Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215--219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affirmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.