Bisymmetric matrix

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title: "Bisymmetric matrix" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["matrices-(mathematics)"] description: "Square matrix symmetric about both its diagonal and anti-diagonal" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Bisymmetric_matrix" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Square matrix symmetric about both its diagonal and anti-diagonal ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/ae/Matrix_symmetry_qtl3.svg" caption="Symmetry pattern of a bisymmetric 5 × 5 matrix"] ::

In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both (it is its own transpose), and , where J is the n × n exchange matrix.

For example, any matrix of the form

\begin{bmatrix} a & b & c & d & e \ b & f & g & h & d \ c & g & i & g & c \ d & h & g & f & b \ e & d & c & b & a \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \ a_{12} & a_{22} & a_{23} & a_{24} & a_{14} \ a_{13} & a_{23} & a_{33} & a_{23} & a_{13} \ a_{14} & a_{24} & a_{23} & a_{22} & a_{12} \ a_{15} & a_{14} & a_{13} & a_{12} & a_{11} \end{bmatrix}

is bisymmetric. The associated 5\times 5 exchange matrix for this example is

J_{5} = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 1 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 \end{bmatrix}

Properties

• Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric.
• The product of two bisymmetric matrices is a centrosymmetric matrix.
• Real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix.{{cite journal |last=Tao |first=David |author2=Yasuda, Mark |title=A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices |journal=SIAM Journal on Matrix Analysis and Applications |volume=23 |issue=3 |pages=885–895 |year=2002 |doi=10.1137/S0895479801386730 |url=https://zenodo.org/record/1236140
• If A is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be bisymmetric.
• The inverse of bisymmetric matrices can be represented by recurrence formulas.

References

References

1. Yasuda, Mark. (2012). "Some properties of commuting and anti-commuting m-involutions". Acta Mathematica Scientia.
2. (2018-01-10). "The inverse of bisymmetric matrices". Linear and Multilinear Algebra.

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