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# Permutation matrix

In linear algebra, a Permutation matrix is a matrix that has exactly one 1 in each row or column and 0s elsewhere. Permutation matrices are the matrix representation of permutations.

For example, the permutation matrix corresponding to σ=(1)(2 4 5 3) is

$P_\sigma = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{bmatrix}$

and

$P_\sigma\begin{bmatrix}g_1\\g_2\\g_3\\g_4\\g_5\end{bmatrix} =\begin{bmatrix}g_1\\g_4\\g_2\\g_5\\g_3\end{bmatrix}.$

In general, for a permutation σ on n objects, the correponding permutation matrix is an n-by-n matrix Pσ is given by Pσ[i,j]=1 if i=σ(j) and 0 otherwise. We have

$P_\sigma\begin{bmatrix}g_1\\ \vdots\\ g_n\end{bmatrix} =\begin{bmatrix}g_{\sigma(1)}\\ \vdots\\ g_{\sigma(n)}\end{bmatrix}$.

Properties:

1. PσPπ=Pσπ for any two permutations σ and π on n objects.
2. P(1) is the identity matrix.
3. Permutation matrices are orthogonal matrix and Pσ-1=Pσ-1.

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