Lower bounds for permanents of Gram matrices having a rank one principal submatrix
The author presents the following: Let \(A=[a_{ij}]\) be a positive semidefinite Hermitian \((n+p)\times (n+p)\) matrix partitioned into blocks \(A_{11}\), \(A_{12}\), \(A_{21}\), and \(A_{22}\) such that \(A_{11}\) is \(n\times n\) and \(A_{22}\) is \(p\times p\). Let \(\tilde A\) denote the \((n+p-1)\times (n+p-1)\) matrix get from A by deleting the first row and column of A. If \(A_{11}\) is of rank one, then \(per(A)\geq na_{11}per(\tilde A)\) with equality iff \(A_{12}\) is the zero matrix or A has a row of zeros. Some examples in this regard are added.
scientific article; zbMATH DE number 4075289