Bisymmetric matrix
Appearance
In
diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = AT (it is its own transpose), and AJ = JA, where J is the n × n exchange matrix
.
For example, any matrix of the form
is bisymmetric. The associated exchange matrix for this example is
Properties
- Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric.
- The product of two bisymmetric matrices is a centrosymmetric matrix.
- 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
- .
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- S2CID 125163794.