Manuals/calci/EXCHANGE

MATRIX("EXCHANGE",order)

$order$ is the order of the Exchange matrix.

Description

This function gives the exchange matrix of order 3.
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.
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

$J_{i,j}={\begin{cases}1,j=n-i+1\\0,j\neq n-i+1\\\end{cases}}$ .

It is also called the reversal matrix,backward identity, or standard involutory permutation.
The form of exchange matrices are

$J_{2}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ $J_{3}={\begin{pmatrix}0&0&1\\0&1&0\\0&0&1\end{pmatrix}}$ $J_{n}={\begin{pmatrix}0&0&\cdots &0&0&1\\0&0&\cdots &0&1&0\\0&0&\cdots &1&0&0\\\vdots &\ddots &\vdots \\0&1&\cdots &0&0&0\\1&0&\cdots &0&0&0\\\end{pmatrix}}$