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 are the same up to sign after pre or post multiplication by the exchange matrix.^[1]

The product of two bisymmetric matrices results in a centrosymetric matrix

References

^ Tao, D. (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. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[simax0-1] Tao, D. (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. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

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