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Permanents of woven matrices

Linear Algebra and its Applications
Publication Date
DOI: 10.1016/s0024-3795(02)00566-9


Abstract A woven matrix, W, is a type of block matrix constructed from an m by n (0,1)-matrix D with row sums r 1, r 2,…, r m and column sums c 1, c 2,…, c n , r i by r i matrices R i ( i=1,2,…, m), and c j by c j matrices, C j ( 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=1 m det R i )(∏ j=1 n det C j ). In this paper it is shown that in general the permanent of W is not determined by the permanents of the R i and C j . However, there are instances when (1) per W=± ∏ i=1 m per R i ∏ j=1 n per C j . For example, it is shown that (I) holds if at least m−1 of the R i 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.

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