Last Updated on September 18, 2025 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.
I have created a YouTube video that covers this post and the next post.
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 2×2 matrix. In the identity matrix, we have 1 in the upper left and the lower right corner, and the remaining corners have 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 the 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 with the first row in the new matrix. Moving the first row of an identity matrix to be the second row and vice versa, then multiplying by 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 about the permutation matrix that it has only one 1 in any of its columns or rows. No more than 3(1s) in the case of a 3×3 matrix. Multiplying the identity matrix by any 3×3 matrix will not cause any changes in the arrangement of rows.
What is a permutation matrix P213?
This is the second arrangement of a permutation matrix (3×3), it is called P213, 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 P213 with a 3×3 matrix is shown in the next slide image.
The content of this post and the next post can be reviewed via the next PDF file.
This is a link to the next post, which will be Permutation matrix-part 2.
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Link to Omni calculator-LU Decomposition Calculator.
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