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Pólya’s permanent problem
 Electron. J. Combin
, 1996
"... A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determ ..."
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A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms. 1
The MoorePenrose inverse of matrices with an acyclic bipartite graph
, 2004
"... The MoorePenrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs. For such a matrix, the sign of every entry of the MoorePenrose inverse is shown to be determined uniquely by the signs of the matrix entries; i.e., the mat ..."
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The MoorePenrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs. For such a matrix, the sign of every entry of the MoorePenrose inverse is shown to be determined uniquely by the signs of the matrix entries; i.e., the matrix has a signed generalized inverse. Necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative MoorePenrose inverse. Nearly reducible matrices are proved to contain no square submatrix having two or more diagonals, implying that a nearly reducible matrix has a signed generalized inverse. Furthermore, it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix.