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On the singular graph and the Weyr characteristic of anM-matrix

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Abstract

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA ii irreducible. LetS be the set of indices α such that the diagonal blockA αα is singular. We define the singular graph ofA to be the setS with partial order defined by α > β if there exists a chain of non-zero blocksA αi, Aij, ⋯, Alβ.

Let Λ1 be the set of maximal elements ofS, and define thep-th levelΛ p ,p = 2, 3, ⋯, inductively as the set of maximal elements ofS \(Λ 1 ⋃ ⋯ ⋃Λ p-1). Denote byλ p the number of elements inΛ p . The Weyr characteristic (associated with 0) ofA is defined to beω (A) = (ω 1, ω2,⋯, ω h ), whereω 1 +⋯ + ω p = dim KerA p,p = 1, 2, ⋯, andω h > 0, ωh+1 = 0.

Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show thatω(A) = (λ 1, ⋯, λh) if and only if certain submatricesD p,p+1 ,p = 1, ⋯, h − 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for −A.

We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS haveω(A) = (λ 1, ⋯, λh). This condition is satisfied ifS is a rooted forest.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, 27607, Raleigh, N.C., USA

    Daniel J. Richman & Hans Schneider

  2. Department of Mathematics, University of Wisconsin, 53707, Madison, Wisconsin, USA

    Daniel J. Richman & Hans Schneider

Authors
  1. Daniel J. Richman
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  2. Hans Schneider
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The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.

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Richman, D.J., Schneider, H. On the singular graph and the Weyr characteristic of anM-matrix. Aequat. Math. 17, 208–234 (1978). https://doi.org/10.1007/BF01818561

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