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The '''permutation matrix''' multiplied by matrix A returns matrix C which is a mirrored transformation of A. A '''permutation matrix''' multiplied by matrix A returns matrix C which is a row-exchanged transformation of A.
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This matrix is simply a diagonal line of 1s in a matrix of 0s, but going the opposite direction as compared to an identity matrix For 3 by 3 matrices, there are 6 possible permutation matrices. They are often denoted based on the rows they exchange, such as P,,2 3,,.
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┌ ┐
│ 0 0 1│
│ 0 1 0│
│ 1 0 0│
└ ┘		 ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│ 1 0 0│ │ 1 0 0│ │ 0 1 0│ │ 0 1 0│ │ 0 0 1│ │ 0 0 1│
│ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│
│ 0 0 1│ │ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│ │ 1 0 0│
└ ┘ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘
(identity matrix) P P (and so on...)
                     2,3 1,2
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│ 0 1│| 1 2| | 3 4|
│ 1 0│| 3 4|=| 1 2|		 │ 0 1││ 1 2│ │ 3 4│
│ 1 0││ 3 4│=│ 1 2│
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| 1 2|│ 0 1│ | 2 1|
| 3 4|│ 1 0│=| 4 3|		 │ 1 2││ 0 1│ │ 2 1│
| 3 4|│ 1 0│=│ 4 3│
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== Inverse Matrices ==

An '''inverse matrix''' is denoted as A^-1^. If a matrix is multiplied by its inverse matrix, it returns the identity matrix.

If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible.

For a square matrix A, the left inverse is the same as the right inverse. AA^-1^ = A^-1^A = I

Special Matrices

Identity Matrix

The identity matrix multiplied by matrix A returns matrix A.

This matrix is simply a diagonal line of 1s in a matrix of 0s. 

┌      ┐
│ 1 0 0│
│ 0 1 0│
│ 0 0 1│
└      ┘

Permutation Matrix

A permutation matrix multiplied by matrix A returns matrix C which is a row-exchanged transformation of A.

For 3 by 3 matrices, there are 6 possible permutation matrices. They are often denoted based on the rows they exchange, such as P2 3. 

┌      ┐          ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐
│ 1 0 0│          │ 1 0 0│ │ 0 1 0│ │ 0 1 0│ │ 0 0 1│ │ 0 0 1│
│ 0 1 0│          │ 0 0 1│ │ 1 0 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│
│ 0 0 1│          │ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│ │ 1 0 0│
└      ┘          └      ┘ └      ┘ └      ┘ └      ┘ └      ┘
(identity matrix)   P        P        (and so on...)
                     2,3      1,2

Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order. 

┌    ┐┌    ┐ ┌    ┐
│ 0 1││ 1 2│ │ 3 4│
│ 1 0││ 3 4│=│ 1 2│
└    ┘└    ┘ └    ┘

┌    ┐┌    ┐ ┌    ┐
│ 1 2││ 0 1│ │ 2 1│
| 3 4|│ 1 0│=│ 4 3│
└    ┘└    ┘ └    ┘

Inverse Matrices

An inverse matrix is denoted as A-1. If a matrix is multiplied by its inverse matrix, it returns the identity matrix.

If A-1 exists, then A is invertible and non-singular. Not all matrices are invertible.

For a square matrix A, the left inverse is the same as the right inverse. AA-1 = A-1A = I

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LinearAlgebra/SpecialMatrices (last edited 2024-01-30 15:45:39 by DominicRicottone)