In mathematics, a bisymmetric matrix is a square matrix symmetric about both diagonals. More precisely, a square n × n matrix A is called bisymmetric if two statements are true:
- A = A T ,
- AJ = JA , where J is the n × n exchange matrix .
If the matrix A has components a i, j , then these statements can be rewritten as:
- a i, j = a j, i ,
- a n + 1-i, n + 1-j = a n + 1-j, n + 1-i .
For example:
Properties
Bisymmetric matrices are both symmetric centrosymmetric matrices and symmetric persymmetric matrices. It was shown that real bisymmetric matrices are those and only those matrices whose eigenvectors do not change up to a sign when multiplied by the exchange matrix . [one]
The product of two bisymmetric matrices is a centrosymmetric matrix .
The number of different elements of the bisymmetric n × n matrix is where through the operation of taking the whole part is indicated.
Notes
- ↑ Tao, D .; Yasuda, M. A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices (English) // SIAM J. Matrix Anal. Appl. : journal. - 2002. - Vol. 23 , no. 3 . - P. 885-895 . - DOI : 10.1137 / S0895479801386730 . (inaccessible link)