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These '''special matrices''' are core concepts to linear algebra. |
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| A '''permutation matrix''' multiplied by matrix A returns a row- or column-exchanged transformation of A, depending on the order of multiplication. | A '''permutation matrix''' multiplied by matrix A returns a row- or column-exchanged transformation of A. If the permutation matrix leads in the multiplication, rows are exchanged. If the permutation matrix follows, columns are exchanged. |
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| See [[LinearAlgebra/PermutationMatrices|Permutation Matrices]] for more information. | The transpose permutation matrix is the same as the inverse permutation matrix: P^T^ = P^-1^. |
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| ---- | The transpose permutation matrix multiplied by the permutation matrix is the same as the identity matrix: P^T^P = I |
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| == Inverse Matrices == | === Counting Permutations === |
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| 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]]. |
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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| julia> A = [1 2; 2 1] 2×2 Matrix{Int64}: 1 2 2 1 julia> A == A' true |
┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ │ 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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| See [[LinearAlgebra/MatrixTransposition#SymmetricMatrices|Symmetric Matrices]] for more information. | For any ''n'' by ''n'' matrix, there are ''n''! possible permutation matrices. |
Special Matrices
These special matrices are core concepts to linear algebra.
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. If the permutation matrix leads in the multiplication, rows are exchanged. If the permutation matrix follows, columns are exchanged.
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 7The transpose permutation matrix is the same as the inverse permutation matrix: PT = P-1.
The transpose permutation matrix multiplied by the permutation matrix is the same as the identity matrix: PTP = I
Counting Permutations
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 1,2For any n by n matrix, there are n! possible permutation matrices.