A relation between the permanental and determinantal adjoints

Abstract

Let Hn denote the set of complex n-square positive semidefinite hermtian matrices. We partially order Hn: If A, B, A–B єHn, write A > B. For A єHn, Write P(A) for the permanental adjoint of A, i.e., P(A) is the n-suare matrix whose i, j entry is per A(j/i), where A(j/i) is the submatrix of A obtained by deleting row j and column i. Now, P(A) is a principal submatrix of t he (n–1)st induced power matrix of AT. Hence, P(A) ∈Hn. Also D(A), the classical adjoing, is in Hn.