Last Updated on January 5, 2024 by Maged kamel

Permutation matrix.

Our subject will be the introduction to the permutation matrix. There is a definition of the permutation matrix that it is an identity matrix with interchanged rows.

With the help of this permutation matrix, we can create a new matrix different from the old matrix, where there will be a change of rows. This is a brief illustration of the permutation matrix quoted from Prof. Mathew W Reid. The author explains that with the change in the process of multiplication, we can get either a swap in rows or a swap in columns.

First, we start with the identity matrix for a matrix of 2×2. In the identity matrix, we have 1 in the upper left and the lower right corner and the remaining corners are having zeros. Multiplying an identity matrix to a matrix 2xc2 will not change the arrangement of rows or columns.

How to swap the first row in a 2×2 matrix?

We have only two options in the 2×2 matrix to swap the rows, the first option is to swap the first row to be as a second row when we multiply a given matrix from the left with a permutation matrix P.

The second option is to swap the second row to the first row in the new matrix. Moving the first row of an identity matrix to be a second row and vice versa, then multiplying to another matrix will create a new matrix with shifted rows. as we can see in the next slide image.

How to swap the first column in a 2×2 matrix?

If we want to change the arrangement of the columns of a matrix, we can multiply this matrix by a permutation matrix, where its first column is (0 1) and its second column is (1 0), this arrangement will produce a new matrix arranged as (b a) as a first column and (d c) as a second column. The original matrix arrangement was (a b) as a first row and ( C d) as a second row. Multiplying any matrix from the right by a permutation matrix ( 0 1, 1 0) will create a swap in the columns.

How many permutation matrices are in a 3×3 matrix?

We have (3x2x1)=6 permutation matrices for the 3×3 matrix. 6 sketches showing these arrangements in the next slide image.

The first option is the normal identity matrix I, where row 1 is followed by row 2 and at the end by row 3. There is an interesting remark as for the permutation matrix that it has only one 1 in any of its columns or rows. No more than 3(1 s) in the case of a 3×3 matrix. Multiplying the identity matrix with any matrix 3×3 will not cause any changes in the arrangement of rows.

What is a permutation matrix P12?

This is the second arrangement of a permutation matrix (3×3) it is called P12, where there is a swap between row 1 to row 2 and accordingly from r2 to r1, while the third row remains unchanged.

Such a matrix when multiplied by any other matrix will create a new matrix where the first row of the new matrix is the second row in the original matrix and the new second row is the first row of the original matrix. A given example of the multiplication of a permutation matrix P12 with a matrix 3×3 is shown in the next slide image.

This is a link for the PDF file used in the illustration of this post and the next post.

This is a link to the next post which will be Permutation matrix-part 2.

This is the Omni calculator for estimating various items of linear algebra -LU Decomposition Calculator.

Link to Omni calculator-LU Decomposition Calculator.

Another calculator to use is Calculator for matrices.