Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A centrosymmetric matrix is a square matrix that commutes with the exchange matrix (the matrix wit..."

2014-05-01T06:38:33Z

Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A centrosymmetric matrix is a square matrix that commutes with the exchange matrix (the matrix wit..."

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{{square matrix property}}

==Definition==

===Verbal definition===

A centrosymmetric matrix is a [[square matrix]] that commutes with the [[exchange matrix]] (the matrix with 1s on the antidiagonal and 0s elsewhere) of the same number of rows and columns.

===Algebraic definition===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> matrix <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is termed a '''centrosymmetric matrix''' if it satisfies the following condition:

<math>a_{ij} = a_{n+1-i,n+1-j} \ \forall i,j \in \{ 1,2,\dots, \}</math>

In symbols, if <math>J</math> denotes the [[exchange matrix]], then <math>A</math> is centrosymmetric if and only if <math>AJ = JA</math>.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::symmetric Toeplitz matrix]] || || || || {{intermediate notions short|centrosymmetric matrix|symmetric Toeplitz matrix}}
|-
| [[Weaker than::bisymmetric matrix]] || centrosymmetric and a [[symmetric matrix]] || || || {{intermediate notions short|centrosymmetric matrix|bisymmetric matrix}}
|}

Centrosymmetric matrix - Revision history

Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A centrosymmetric matrix is a square matrix that commutes with the exchange matrix (the matrix wit..."