Symmetric, Hankel-Symmetric, and Centrosymmetric Doubly Stochastic Matrices


We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel-symmetric, centrosymmetric, and both symmetric and Hankel-symmetric. We determine dimensions of these polytopes and classify their extreme points. We also determine a basis of the real vector spaces generated by permutation matrices with these special structures.

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  1. 1.

    We use the superscript ‘t’ to reflect the fact that symmetric matrices are invariant under transposition.

  2. 2.

    The motivation for calling this the Hankel-diagonal was that Hankel matrices are constant on the antidiagonal and on all diagonals parallel to it. In contrast, Toeplitz matrices are matrices constant on the main diagonal and all diagonals parallel to it. We use Ah to denote the matrix obtained from a square matrix A by reflection about the Hankel-diagonal and are tempted to think of the main diagonal as the Toeplitz diagonal and the ‘t’ in At to stand for Toeplitz.

  3. 3.

    Also called persymmetric but we prefer Hankel-symmetric.

  4. 4.

    Sometimes also called doubly symmetric.

  5. 5.

    Note that the graphs G(A) and Gh(A) are not considered as multigraphs; at most one edge (corresponding to a nonzero entry) joins a pair of vertices. This is in contrast to the bipartite multigraph BG(A) in which more than one edge can join a pair of vertices, equivalently, edges have weights assigned to them.


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Correspondence to Richard A. Brualdi.

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Brualdi, R.A., Cao, L. Symmetric, Hankel-Symmetric, and Centrosymmetric Doubly Stochastic Matrices. Acta Math Vietnam 43, 675–700 (2018).

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  • Matrix
  • Permutation matrix
  • Symmetric
  • Hankel-symmetric
  • Centrosymmetric
  • Doubly stochastic
  • Extreme point

Mathematics Subject Classification (2010)

  • 05C50
  • 15B05
  • 15B51
  • 15B48
  • 90C57