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 = A^T and AJ = JA where J is the n × n exchange matrix.

For example:

{\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}}.

Properties

Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric. It has been shown that 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]

The product of two bisymmetric matrices results in a centrosymmetric matrix.

The inverse of bisymmetric matrices can be represented by recurrence formulas.^[2]

References

^ Tao, D.; Yasuda, M. (2002). "A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices". SIAM J. Matrix Anal. Appl. 23 (3): 885–895. doi:10.1137/S0895479801386730. Retrieved 2007-10-12.
^ Wang, Yanfeng; Lü, Feng; Lü, Weiran (2018-01-10). "The inverse of bisymmetric matrices". Linear and Multilinear Algebra. 0 (0): 1–11. doi:10.1080/03081087.2017.1422688. ISSN 0308-1087.

[simax0-1] Tao, D.; Yasuda, M. (2002). "A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices". SIAM J. Matrix Anal. Appl. 23 (3): 885–895. doi:10.1137/S0895479801386730. Retrieved 2007-10-12.

[2] Wang, Yanfeng; Lü, Feng; Lü, Weiran (2018-01-10). "The inverse of bisymmetric matrices". Linear and Multilinear Algebra. 0 (0): 1–11. doi:10.1080/03081087.2017.1422688. ISSN 0308-1087.

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