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== Permutation Matrix == == Permutation Matrices ==
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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 P,,2 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.		 A '''permutation matrix''' multiplied by matrix A returns a row-exchanged transformation of A.
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}}}
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┌ ┐┌ ┐ ┌ ┐
│ 1 2││ 0 1│ │ 2 1│
| 3 4|│ 1 0│=│ 4 3│
└ ┘└ ┘ └ ┘
}}}		 See [[LinearAlgebra/PermutationMatrices|Permutation Matrices]] for more information.
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An '''inverse matrix''' is denoted as A^-1^. If a matrix is multiplied by its inverse matrix, it returns the identity matrix. An '''inverse matrix''' A^-1^ multiplied by matrix A returns the identity matrix.
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For a square matrix A, the left inverse is the same as the right inverse. AA^-1^ = A^-1^A = I See [[LinearAlgebra/MatrixInversion|Matrix Inversion]] for more information.



== Symmetric Matrices ==

A '''symmetric matrix''' is any matrix that is equal to its [[LinearAlgebra/MatrixTransposition|transpose]].

{{{
┌ ┐
│ 1 7│
│ 7 2│
└ ┘
}}}

See [[LinearAlgebra/MatrixTransposition#SymmetricMatrices|Symmetric Matrices]] for more information.

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 Matrices

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

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

See Permutation Matrices for more information.

Inverse Matrices

An inverse matrix A-1 multiplied by matrix A returns the identity matrix.

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

See Matrix Inversion for more information.

Symmetric Matrices

A symmetric matrix is any matrix that is equal to its transpose. 

┌    ┐
│ 1 7│
│ 7 2│
└    ┘

See Symmetric Matrices for more information.

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