Exchange matrix

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In mathematics, especially linear algebra, the exchange matrix is a special case of a permutation matrix, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix.



J_{2}=\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix};\quad J_{3}=\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix};
\quad J_{n}=\begin{pmatrix}
  0      & 0      & \cdots & 0      & 0      & 1      \\
  0      & 0      & \cdots & 0      & 1      & 0      \\
  0      & 0      & \cdots & 1      & 0      & 0      \\
  \vdots & \vdots &        & \vdots & \vdots & \vdots \\ 
  0      & 1      & \cdots & 0      & 0      & 0      \\
  1      & 0      & \cdots & 0      & 0      & 0      
\end{pmatrix}.


[edit] Definition

If J is an n×n exchange matrix, then the elements of J are defined such that:

J_{i,j} = \begin{cases} 
1, & j = n - i \\
0, & j \ne n - i \\
\end{cases}

[edit] Properties

  • JT = J.
  • Jn = I for even n; Jn = J for odd n, where n is any integer. Thus J is an involutary matrix; that is, J−1 = J.
  • The trace of J is 1 if n is odd, and 0 if n is even.

[edit] Relationships

  • Any matrix A satisfying the condition AJ = JA is said to be centrosymmetric.
  • Any matrix A satisfying the condition AJ = JAT is said to be persymmetric.
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