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2 Standard Forms and the Singular Graph
3 Construction of an basis for EA
4 Relationship between uA and an matrix for A
4 other sections not shown
a e j a-th column associated scalars basis for E(A block column ccrA combinatorially dependent Corollary 7.2 Daniel James Richman define denote diagonal blocks dim E(A disjoint subsets $1 eigenspace E(A eigenvalue elementary divisor structure equivalence classes exist non-empty disjoint finite partially ordered following are equivalent Frobenius 1912 full column rank Hence identity matrix inductive assumption irreducible M-matrices Jordan basis Ker AP Lemma Let 1 _ Let a e Let v e linearly dependent linearly ordered lower block triangular M-matrix with singular matrix associated maximal elements non-empty disjoint subsets number of elements Observe partially ordered set permutation matrix positive vectors prove Q.E.D. Remark result follows Rothblum satisfies condition 5.6ii Schneider 1956 semi-positive set with level singular graph singular matrix standard form submatrix thesis University of Wisconsin-Madison Vf e Ker Weyr characteristic