A graph theoretic approach to matrix inversion by partitioning

LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P^–1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D^* ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD^* from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA^* be the adjacency matrix ofD^*. It is easy to show that there exists a permutation matrixQ such thatQA^*Q^–1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.This was an informal talk at the International Symposium on Matrix Computation sponsored by SIAM and held in Gatlinburg, Tennessee, April 24–28, 1961 and was an invited address at the SIAM meeting in Stillwater, Oklahoma on August 31, 1961