Some Combinatorially Defined Matrix Classes
In this section we consider the symmetric group of permutations of a finite set and their partial order known as the Bruhat order. Regarding a permutation as a permutation matrix, this partial order is related to Gaussian elimination and leads to the matrix Bruhat decomposition of a nonsingular matrix, and then to a characterization of ags in a vector space. We also describe a correspondence between permutations that are involutions (symmetric permutation matrices) and a certain class of nonnegative integral matrices.
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