A graph theoretic approach to matrix inversion by partitioning
 Frank Harary
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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.
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 Title
 A graph theoretic approach to matrix inversion by partitioning
 Journal

Numerische Mathematik
Volume 4, Issue 1 , pp 128135
 Cover Date
 19621201
 DOI
 10.1007/BF01386304
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 SpringerVerlag
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 Authors

 Frank Harary ^{(1)}
 Author Affiliations

 1. Mathematics Department, The University of Michigan, Ann Arbor, Michigan