Inequalities for permanents involving Perron complements

Abstract

Let A∈ R n,n and let α and β be nonempty complementary subsets of {1,…, n} of increasing integers. For λ> ρ( A[ β]), we define the generalized Perron complement of A[ β] in A at λ as the matrix P λ(A/A[β])=A[α]+A[α,β](λI−A[β]) −1A[β,α] . For the classes of the nonnegative matrices and of the positive semidefinite matrices, we study the relationship between the permanents of the whole matrices and the permanents of their Perron complement. Our conditions, which hold in many cases of interest, are such that the value of the permanent increases as we pass from the whole matrix to its generalized Perron complement.For nonnegative and irreducible matrices, we also study the relationship between the maximum circuit geometric mean of the entire matrix and the maximum circuit geometric mean of its Perron complements.