On permanental compounds

If k⩽n, then Gk,n denotes the set of all strictly increasing functions from {1,2,…,k} to {1,2,…,n} ordered lexicographically. If A=[aij] is an n×n complex matrix, then A[α∣β] denotes the k×k submatrix of A whose rows and columns are specified by α and β, respectively, and A(α∣β) is the corresponding complementary submatrix of A. The k-th permanental compound, CPk(A), of A is the matrix indexed by the members of Gk,n such that (CPk(A))α,β=per(A[α∣β]) for all α,β∈Gk,n. Much work has been done on permanental compounds. We consider the matrix Ck(A) such that (Ck(A))α,β=per(A(α∣β))per(A[α∣β]). In 1986 Bapat and Sunder [1] conjectured that if A is positive semi-definite and Hermitian, then per(A) is the largest eigenvalue C1(A). We extend the conjecture to all of the matrices Ck(A) where 1⩽k⩽n. We show how our extended conjecture is related to a theorem of this author, and derive several inequalities. For example, we show that if x is a (0,1)-vector, then 〈Ck(A)x,x〉⩽‖x‖2per(A) for all positive semi-definite A.