Permutation Matrices

1. Permutation Matrices are Orthogonal Square Matrices that are formed by Re-Arranging the Columns and/or Rows of Identity Matrix and are used to Re-Arrange the Columns and/or Rows of any other given Matrix. Following are some examples of Permutation Matrices
   
   \(\begin{bmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\1& 0 & 0 & 0 \\0 & 1 & 0 & 0\end{bmatrix}\hspace{5mm}\begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix}\)
   
   Just like any other Orthogonal Square Matrix, the Inverse of any Permutation Matrix is same as its Transpose.
2. The Rows of any \(M \times N\) Matrix \(A\) can be Rearranged by Pre-Multiplying the Matrix \(A\) with any \(M \times M\) Permutation Matrix \(P\).
3. The Columns of any \(M \times N\) Matrix \(A\) can be Rearranged by Post-Multiplying the Matrix \(A\) with any \(N \times N\) Permutation Matrix \(P\).
4. To demonstrate how the Permutation Matrices work, let's consider a \(4 \times 4\) Matrix \(A\) having elements \(a_{ij}\) and a \(4 \times 4\) Permutation Matrix \(P\) having elements \(p_{ij}\) as given below
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44}\end{bmatrix} \hspace{.5cm}P=\begin{bmatrix} 0 & p_{12} & 0 & 0\\ p_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & p_{34} \\ 0 & 0 & p_{43} & 0\end{bmatrix}\)
   
   Pre-Multiplying the Permutation Matrix \(P\) with Matrix \(A\) Re-Arranges Rows of the Matrix \(A\)as given in the following
   
   \(PA=\begin{bmatrix} 0 & p_{12} & 0 & 0\\ p_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & p_{34} \\ 0 & 0 & p_{43} & 0\end{bmatrix}\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44}\end{bmatrix}= \begin{bmatrix} p_{12} \cdot a_{21} & p_{12} \cdot a_{22} & p_{12} \cdot a_{23} & p_{12} \cdot a_{24} \\ p_{21} \cdot a_{11} & p_{21} \cdot a_{12} & p_{21} \cdot a_{13} & p_{21} \cdot a_{14}\\ p_{34} \cdot a_{41} & p_{34} \cdot a_{42} & p_{34} \cdot a_{43} & p_{34} \cdot a_{44}\\ p_{43} \cdot a_{31} & p_{43} \cdot a_{32} & p_{43} \cdot a_{33} & p_{43} \cdot a_{34} \end{bmatrix}\)
   
   Please note that The Unity Element of \(i^{th}\) Row and \(j^{th}\) Column of the Permutation Matrix \(P\) Gets Multiplied to the \(j^{th}\) Row of Matrix \(A\) and the \(j^{th}\) Row of Matrix \(A\) gets Re-Arranged to \(i^{th}\) Row of Matrix \(A\).
   
   Post-Multiplying the Permutation Matrix \(P\) with Matrix \(A\) Re-Arranges Columns of the Matrix \(A\) as given in the following
   
   \(AP=\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44}\end{bmatrix}\begin{bmatrix} 0 & p_{12} & 0 & 0\\ p_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & p_{34} \\ 0 & 0 & p_{43} & 0\end{bmatrix}= \begin{bmatrix} p_{21} \cdot a_{12} & p_{12} \cdot a_{11} & p_{43} \cdot a_{14} & p_{34} \cdot a_{13} \\ p_{21} \cdot a_{22} & p_{12} \cdot a_{21} & p_{43} \cdot a_{24} & p_{34} \cdot a_{23} \\ p_{21} \cdot a_{32} & p_{12} \cdot a_{31} & p_{43} \cdot a_{34} & p_{34} \cdot a_{33} \\ p_{21} \cdot a_{42} & p_{12} \cdot a_{41} & p_{43} \cdot a_{44} & p_{34} \cdot a_{43} \end{bmatrix}\)
   
   Please note that The Unity Element of \(i^{th}\) Row and \(j^{th}\) Column of the Permutation Matrix \(P\) Gets Multiplied to the \(i^{th}\) Column of Matrix \(A\) and the \(i^{th}\) Column of Matrix \(A\) gets Re-Arranged to \(j^{th}\) Column of Matrix \(A\).
5. All Permutation Matrices are Periodic Matrices.