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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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| A '''permutation matrix''' multiplied by matrix A returns a row-exchanged transformation of A. | 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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| ┌ ┐┌ ┐ ┌ ┐ │ 0 1││ 1 2│ │ 3 4│ │ 1 0││ 3 4│=│ 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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| ┌ ┐ │ 1 7│ │ 7 2│ └ ┘ |
julia> A = [1 2; 2 1] 2×2 Matrix{Int64}: 1 2 2 1 julia> A == A' true |
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.