Bisymmetric 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 A = AT and AJ = JA where J is the n × n exchange matrix.

Symmetry pattern of a bisymmetric 5×5 matrix

For example:

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.[1]
  • If A is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be bisymmetric.[2]
  • The inverse of bisymmetric matrices can be represented by recurrence formulas.[3]

References
  1. Tao, David; Yasuda, Mark (2002). "A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices". SIAM Journal on Matrix Analysis and Applications. 23 (3): 885–895. doi:10.1137/S0895479801386730.
  2. Yasuda, Mark (2012). "Some properties of commuting and anti-commuting m-involutions". Acta Mathematica Scientia. 32 (2): 631–644. doi:10.1016/S0252-9602(12)60044-7.
  3. Wang, Yanfeng; Lü, Feng; Lü, Weiran (2018-01-10). "The inverse of bisymmetric matrices". Linear and Multilinear Algebra. 0 (3): 479–489. doi:10.1080/03081087.2017.1422688. ISSN 0308-1087.
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