Abstract :
If kless-than-or-equals, slantn, 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[αmidβ] denotes the k×k submatrix of A whose rows and columns are specified by α and β, respectively, and A(αmidβ) is the corresponding complementary submatrix of A. The k-th permanental compound, CPk(A), of A is the image matrix indexed by the members of Gk,n such that (CPk(A))α,β=per(A[αmidβ]) for all α,βset membership, variantGk,n. Much work has been done on permanental compounds. We consider the image matrix image such that image. In 1986 Bapat and Sunder [1] conjectured that if A is positive semi-definite and Hermitian, then per(A) is the largest eigenvalue image. We extend the conjecture to all of the matrices image where 1less-than-or-equals, slantkless-than-or-equals, slantn. 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 image for all positive semi-definite A.
