Permutation Matrix

A permutation matrix is a square matrix. It’s always raw equivalent to the Identity matrix

For size $n$ there exists $n!$ permutation matrcies.

For example there exists $2$ permutation matrcies of size $2$ which are:

A permutation matrix is non-singular, and the determinant is always $\pm 1$

You can imagine the 1’s in the $n \times n$ matrix as non attacking rooks in an $n \times n$ chessboard

Permutation In Matrix Matrix Multiplication

Left Multiplication

Left Multiplication by a permutation matrix rearranges the corresponding rows

Let's See some julia Code

A = [[0, 1 , 0] [0, 0, 1] [1, 0, 0]]
B = [[1, 2, 3] [4, 5, 6] [7, 8, 9]]
@show A * B

3×3 Matrix{Int64}:
 3  6  9
 1  4  7
 2  5  8

Right Multiplication

Right multiplication by a permutation matrix rearranges the corresponding columns

Let's test with julia

A = [[0, 1 , 0] [0, 0, 1] [1, 0, 0]]
B = [[1, 2, 3] [4, 5, 6] [7, 8, 9]]
@show B * A

3×3 Matrix{Int64}:
 4  7  1
 5  8  2
 6  9  3

We can also combine both as follows

I found This matrix very interesting! It’s also an orthogonal matrix , which means it’s equal to its transpose. What about you? Do also you find it interesting?