A note on permanents and generalized complementary basic matrices

Abstract

We say that the product of a row vector and a column vector is intrinsic if there is at most one non-zero product of corresponding coordinates. Analogously we speak about intrinsic product of two or more matrices, as well as about intrinsic factorizations of matrices. Since all entries of the intrinsic product are products of entries of the multiplied matrices, there is no addition. The class of complementary basic matrices (CB-matrices) was recently introduced as matrices, if of order n, A=Gi1Gi2⋯Gin-1, where (i1,i2,…,in-1) is a permutation of (1,2,…,n-1), and the matrices Gk, k=1,…,n-1 have the formGk=Ik-1CkIn-k-1for some 2×2 matrices Ck. It was observed that (1). independently of the permutation, all such matrices with given Ci’s have the same spectrum (though they do not form a similarity class), (2) the classical companion matrix belongs to the class of CB-matrices, (3) the multiplication of the Gi’s is intrinsic. Connections between the 2×2 matrices Ck and the resulting CB-matrix ΠGk have been explored, in particular, the properties which are inherited from the Ck to the ∏Gk. Two situations were considered, for the ordinary real CB-matrices and for the corresponding sign pattern matrices. In this paper, we consider generalized complementary basic matrices, where the Ck matrices are replaced by square matrices of arbitrary sizes. Some interesting facts and results about the permanents and permanental polynomials of such matrices are obtained.