Centrosymmetric matrix

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 = [ Ai,j ] is centrosymmetric when its entries satisfy

Ai,j = An−i+1,n−j+1 for 1 ≤ i,j ≤ n.
Symmetry pattern of a centrosymmetric 5×5 matrix

If J denotes the n × n matrix with 1 on the counterdiagonal and 0 elsewhere (that is, Ji,n+1-i = 1; Ji,j = 0 if j ≠ n+1-i), then a matrix A is centrosymmetric if and only if AJ = JA. The matrix J is sometimes referred to as the exchange matrix.

Examples
  • All 2×2 centrosymmetric matrices have the form
  • All 3×3 centrosymmetric matrices have the form

Algebraic structure and properties
  • If A and B are centrosymmetric matrices over a given field F, then so are A+B and cA for any c in F. 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 F is a subalgebra of the associative algebra of all n × n matrices.
  • If A is a centrosymmetric matrix with an m-dimensional eigenbasis, then its m eigenvectors can each be chosen so that they satisfy either x=Jx or x=-Jx.
  • If A is a centrosymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be centrosymmetric.[1]

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

The centrosymmetric relation AJ = JA lends itself to a natural generalization, where J is replaced with an involutory matrix K (i.e., K2 = I)[2] [3] [4] or, more generally, a matrix K satisfying Km = I for an integer m > 1.[1] The inverse problem for the commutation relation AK = KA of identifying all involutory K that commute with a fixed matrix A, has also been studied.[1]

Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field is the field of real numbers, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre or post multiplication by the exchange matrix.[3] A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.[5]

References
  1. Yasuda, Mark (2012). "Some properties of commuting and anti-commuting m-involutions". Acta Mathematica Scientia. 32 (2): 631–644. doi:10.1016/S0252-9602(12)60044-7.
  2. Andrew, Alan (1973). "Eigenvectors of certain matrices". Linear Algebra Appl. 7 (2): 151–162. doi:10.1016/0024-3795(73)90049-9.
  3. Tao, David; Yasuda, Mark (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.
  4. Trench, W. F. (2004). "Characterization and properties of matrices with generalized symmetry or skew symmetry". Linear Algebra Appl. 377: 207–218. doi:10.1016/j.laa.2003.07.013.
  5. Yasuda, Mark (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". SIAM J. Matrix Anal. Appl. 25 (3): 601–605. doi:10.1137/S0895479802418835.

Further reading
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