Permanents of woven matrices Original Research Article

A woven matrix, W, is a type of block matrix constructed from an m by n (0,1)-matrix D with row sums r1,r2,…,rm and column sums c1,c2,…,cn, ri by ri matrices Ri (i=1,2,…,m), and cj by cj matrices, Cj (j=1,2,…,n). Several properties of the determinant and the spectrum of woven matrices are known. In particular, the determinant of a woven matrix is ±(∏i=1mdetRi)(∏j=1ndetCj). In this paper it is shown that in general the permanent of W is not determined by the permanents of the Ri and Cj. However, there are instances whenimageFor example, it is shown that (I) holds if at least m−1 of the Ri are diagonal matrices. The main result of the paper is a characterization of the D’s for which each woven matrix, W, using D satisfies (I). As an application, we determine families of matrices whose permanents can be efficiently computed using determinants.