Differences between revisions 1 and 5 (spanning 4 versions)
⇤ ← Revision 1 as of 2021-09-14 15:49:24 →
Size: 1019
Editor: DominicRicottone
Comment:
← Revision 5 as of 2022-03-19 20:45:13 → ⇥
Size: 1268
Editor: DominicRicottone
Comment:
Deletions are marked like this. Additions are marked like this.
Line 19: Line 19:
== Permutation Matrix == == Permutation Matrices ==
Line 21: Line 21:
The '''permutation matrix''' multiplied by matrix A returns matrix C which is a mirrored transformation of A.

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

{{{
┌ ┐
│ 0 0 1│
│ 0 1 0│
│ 1 0 0│
└ ┘
}}}

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.
Line 37: Line 25:
│ 0 1│| 1 2| | 3 4|
│ 1 0│| 3 4|=| 1 2|
└ ┘└ ┘ └ ┘

┌ ┐┌ ┐ ┌ ┐
| 1 2|│ 0 1│ | 2 1|
| 3 4|│ 1 0│=| 4 3|		 │ 0 1││ 1 2│ │ 3 4│
│ 1 0││ 3 4│=│ 1 2│
Line 46: Line 29:

See [[LinearAlgebra/PermutationMatrices|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 [[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.

CategoryRicottone

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