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<<TableOfContents>> ---- |
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| The '''identity matrix''' multiplied by matrix A returns matrix A. | The '''identity matrix''' is a diagonal line of 1s in a matrix of 0s. |
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| This matrix is simply a diagonal line of 1s in a matrix of 0s. | Any matrix A multiplied by the (appropriately sized) identity matrix returns matrix A. |
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| ┌ ┐ │ 1 0 0│ │ 0 1 0│ │ 0 0 1│ └ ┘ |
julia> using LinearAlgebra julia> Matrix{Int8}(I,3,3) 3×3 Matrix{Int8}: 1 0 0 0 1 0 0 0 1 |
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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,,. |
A '''permutation matrix''' multiplied by matrix A returns a row- or column-exchanged transformation of A, depending on the order of multiplication. |
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| ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ │ 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 |
julia> P = Matrix{Int8}(I,3,3)[:,[3,2,1]] 3×3 Matrix{Int8}: 0 0 1 0 1 0 1 0 0 julia> A = [1 2 3; 4 5 6; 7 8 9] 3×3 Matrix{Int64}: 1 2 3 4 5 6 7 8 9 julia> P * A 3×3 Matrix{Int64}: 7 8 9 4 5 6 1 2 3 julia> A * P 3×3 Matrix{Int64}: 3 2 1 6 5 4 9 8 7 |
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| Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order. | See [[LinearAlgebra/PermutationMatrices|Permutation Matrices]] for more information. |
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| {{{ ┌ ┐┌ ┐ ┌ ┐ │ 0 1││ 1 2│ │ 3 4│ │ 1 0││ 3 4│=│ 1 2│ └ ┘└ ┘ └ ┘ ┌ ┐┌ ┐ ┌ ┐ │ 1 2││ 0 1│ │ 2 1│ | 3 4|│ 1 0│=│ 4 3│ └ ┘└ ┘ └ ┘ }}} |
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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]]. {{{ julia> A = [1 2; 2 1] 2×2 Matrix{Int64}: 1 2 2 1 julia> A == A' true }}} See [[LinearAlgebra/MatrixTransposition#SymmetricMatrices|Symmetric Matrices]] for more information. |
Special Matrices
Identity Matrix
The identity matrix is a diagonal line of 1s in a matrix of 0s.
Any matrix A multiplied by the (appropriately sized) identity matrix returns matrix A.
julia> using LinearAlgebra
julia> Matrix{Int8}(I,3,3)
3×3 Matrix{Int8}:
1 0 0
0 1 0
0 0 1
Permutation Matrices
A permutation matrix multiplied by matrix A returns a row- or column-exchanged transformation of A, depending on the order of multiplication.
julia> P = Matrix{Int8}(I,3,3)[:,[3,2,1]]
3×3 Matrix{Int8}:
0 0 1
0 1 0
1 0 0
julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Matrix{Int64}:
1 2 3
4 5 6
7 8 9
julia> P * A
3×3 Matrix{Int64}:
7 8 9
4 5 6
1 2 3
julia> A * P
3×3 Matrix{Int64}:
3 2 1
6 5 4
9 8 7See 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.
julia> A = [1 2; 2 1]
2×2 Matrix{Int64}:
1 2
2 1
julia> A == A'
trueSee Symmetric Matrices for more information.