Determinantal inequalities for diagonally signed matrices and an application to Gram-Cauchy matrices

AbstractThe condition that a Hermitian matrix is diagonally signed (complementary) has recently been shown to guarantee that its signature is invariant with respect to Hadamard products with Gram matrices. In this paper we establish inequalities for the determinants of these diagonally signed matrices that are analogs of well-known inequalities for positive definite matrices. Because Hermitian Cauchy matrices and their confluent forms are diagonally signed, we can then infer from the new inequalities the existence (in general) of inverses of the confluent forms of Hermitian Gram-Cauchy matrices