Centrosymmetric matrix

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In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n × n matrix A = [ A_i,j ] is centrosymmetric if and only if its entries satisfy A_i,j = A_{n−i+1,n−j+1} for 1 ≤ i,j ≤ n. If J denotes the n × n matrix with 1's on the counterdiagonal and 0's elsewhere (that is, J_{i,n − i} = 1; J_i,j = 0 if j ≠ n − i), then a matrix A is centrosymmetric if and only if AJ = JA. The matrix J is sometimes referred to as the exchange matrix.

1 Examples
2 Algebraic structure
3 Related structures
4 Footnotes
5 References
6 External links

[edit] Examples

All 2×2 centrosymmetric matrices have the form

$\begin{bmatrix} a & b \\ b & a \end{bmatrix}.$

All 3×3 centrosymmetric matrices have the form

$\begin{bmatrix} a & b & c \\ d & e & d \\ c & b & a \end{bmatrix}.$

Symmetric Toeplitz matrices are centrosymmetric.

[edit] Algebraic structure

If A and B are centrosymmetric matrices over a given field K, then so are A+B and cA for any c in K. In addition, the matrix product AB is centrosymmetric, since JAB = AJB = ABJ. Since the identity matrix is also centrosymmetric, it follows that the set of n × n centrosymmetric matrices over K is a subalgebra of the associative algebra of all n × n matrices.

[edit] Related structures

An n × n matrix A is said to be skew-centrosymmetric if and only if its entries satisfy A_i,j = -A_{n−i+1,n−j+1} for 1 ≤ i,j ≤ n. Equivalently, A is skew-centrosymmetric if and only if AJ = -JA, where J is the exchange matrix defined above.

The centrosymmetric relation AJ = JA lends itself to a natural generalization (see e.g. ^[1] ^[2] ^[3]) , where J is replaced with an involutory matrix K (i.e., K² = I).

[edit] Footnotes

^ A. Andrew, Eigenvectors of certain matrices, Linear Algebra Appl., 7 (1973), pp. 151–162.
^ D. Tao and M. Yasuda, A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 885–895.
^ W.F. Trench, Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear Algebra Appl. 377 (2004) 207–218.

[edit] References

T. Muir (1960), A Treatise on the Theory of Determinants, Dover, p. 19. ISBN 0-486-60670-8.
J. R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, American Mathematical Monthly 92 (1985), pp. 711-717.