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*The exchange matrix is the square matrix of a permutation matrix.
*The exchange matrix is the square matrix of a permutation matrix.
*In this matrix the 1 elements reside on the counterdiagonal and all other elements are zero.
*In this matrix the 1 elements reside on the counterdiagonal and all other elements are zero.
*It is a 'row-reversed' or 'column-reversed' version of the identity matrix.
*It is a 'row-reversed' or 'column-reversed' version of the identity matrix.
* Suppose J is an nxn exchange matrix, then the elements of J are defined such that
<math> J_{i,j}=\begin{cases}
<math> J_{i,j}=\begin{cases}
1, j=n-i+1 \\
1, j=n-i+1 \\
\end{cases}</math>.
\end{cases}</math>.
*It is also called the reversal matrix,backward identity, or standard involutory permutation.
*It is also called the reversal matrix,backward identity, or standard involutory permutation.
*The form of exchange matrices are J2=();J3=(); and so on.
*The form of exchange matrices are
<math>J_2=\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}</math>
<math>J_3=\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}</math>()
<math> J_n =\begin{pmatrix}
0 & 0 & \cdots & 0 & 0 & 1 \\
0 & 0 & \cdots & 0 & 1 & 0 \\
0 & 0 & \cdots & 1 & 0 & 0 \\
\vdots & \ddots & \vdots \\
0 & 1 & \cdots & 0 & 0 & 0 \\
1 & 0 & \cdots & 0 & 0 & 0 \\
\end{pmatrix}</math>