Bisymmetric matrix: Difference between revisions
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==Properties== |
==Properties== |
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Bisymmetric matrices are both symmetric [[centrosymmetric matrix|centrosymmetric]] and symmetric [[persymmetric matrix|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.<ref name="simax0">{{cite journal | last = Tao | first = D. | |
Bisymmetric matrices are both symmetric [[centrosymmetric matrix|centrosymmetric]] and symmetric [[persymmetric matrix|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.<ref name="simax0">{{cite journal | last = Tao | first = D. |author2=Yasuda, M. | title = A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices | journal = SIAM J. Matrix Anal. Appl. | volume = 23 | issue = 3 | pages = 885–895 | year = 2002 | url = http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJMAEL000023000003000885000001&idtype=cvips&gifs=Yes |
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| accessdate = 2007-10-12 | doi = 10.1137/S0895479801386730}}</ref> |
| accessdate = 2007-10-12 | doi = 10.1137/S0895479801386730}}</ref> |
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Revision as of 12:03, 10 April 2014
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.
For example:
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 centrosymmetric matrix
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.