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These '''special matrices''' are core concepts to linear algebra. <<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. | 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 7 }}} The transpose permutation matrix is the same as the inverse permutation matrix: P^T^ = P^-1^. The transpose permutation matrix multiplied by the permutation matrix is the same as the identity matrix: P^T^P = I === Counting Permutations === |
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| Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order. {{{ ┌ ┐┌ ┐ ┌ ┐ │ 0 1││ 1 2│ │ 3 4│ │ 1 0││ 3 4│=│ 1 2│ └ ┘└ ┘ └ ┘ ┌ ┐┌ ┐ ┌ ┐ │ 1 2││ 0 1│ │ 2 1│ | 3 4|│ 1 0│=│ 4 3│ └ ┘└ ┘ └ ┘ }}} == Inverse Matrices == An '''inverse matrix''' is denoted as A^-1^. If a matrix is multiplied by its inverse matrix, it returns the identity matrix. If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible. For a square matrix A, the left inverse is the same as the right inverse. AA^-1^ = A^-1^A = I |
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.