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| The '''identity matrix''' is a diagonal line of 1s in a matrix of 0s. | The '''identity matrix''' is a diagonal line of ones in a matrix of zeros. |
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| Any matrix A multiplied by the (appropriately sized) identity matrix returns matrix A. | Any matrix '''''A''''' multiplied by the (appropriately sized) identity matrix returns matrix '''''A'''''. |
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| 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. | 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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| The transpose permutation matrix is the same as the inverse permutation matrix: P^T^ = P^-1^. | 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 | The transpose permutation matrix multiplied by the permutation matrix is the same as the identity matrix: '''''P'''^T^'''P''' = '''I''''' |
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| 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,,. | 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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| ---- == Upper Triangular Matrices == If a square matrix has only zeros below the diagonal, it is an '''upper triangular matrix'''. [[LinearAlgebra/Elimination|Gauss-Jordan elimination]] results in a '''row echelon form''' of '''''A''''' which, if '''''A''''' is square, is also upper triangular. [[LinearAlgebra/Orthonormalization|Gram-Schmidt orthonormalization]] is characterized as '''''A = QR''''' where '''''R''''' is upper triangular. ---- == Lower Triangular Matrices == If a square matrix has only zeros above the diagonal, it is a '''lower triangular matrix'''. If [[LinearAlgebra/Elimination|Gauss-Jordan elimination]] is continued into backwards elimination, it results in a '''reduced row echelon form''' of '''''A'''''. If '''''A''''' is square, it will be both upper and lower triangular. |
Special Matrices
These special matrices are core concepts to linear algebra.
Contents
Identity Matrix
The identity matrix is a diagonal line of ones in a matrix of zeros.
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.
Upper Triangular Matrices
If a square matrix has only zeros below the diagonal, it is an upper triangular matrix.
Gauss-Jordan elimination results in a row echelon form of A which, if A is square, is also upper triangular.
Gram-Schmidt orthonormalization is characterized as A = QR where R is upper triangular.
Lower Triangular Matrices
If a square matrix has only zeros above the diagonal, it is a lower triangular matrix.
If Gauss-Jordan elimination is continued into backwards elimination, it results in a reduced row echelon form of A. If A is square, it will be both upper and lower triangular.