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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 ones in a square matrix of zeros. |
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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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---- |
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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''' is a square matrix of zeros with a one in each row. Multiplying a permutation matrix by some matrix '''''A''''' ('''''PA''''') results in a row-exchanged '''''A'''''. Multiplying some matrix '''''A''''' by a permutation matrix ('''''AP''''') results in a column-exhanged '''''A'''''. |
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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,,. | {{{ 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 === 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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| Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order. | For any ''n'' by ''n'' matrix, there are ''n''! possible permutation matrices. |
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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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| == Inverse Matrices == | == Upper Triangular Matrices == |
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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. | If a square matrix has only zeros below the diagonal, it is an '''upper triangular matrix'''. |
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| If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible. | [[LinearAlgebra/Elimination|Gauss-Jordan elimination]] results in a '''row echelon form''' of '''''A''''' which, if '''''A''''' is square, is also upper triangular. |
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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 | [[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. ---- == Symmetric Matrices == A '''symmetric matrix''' is equal to its [[LinearAlgebra/Transposition|transpose]]. Only a square matrix can be symmetric. These matrices have several useful properties: * The [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are always real. * The [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] can be chosen to be [[LinearAlgebra/Orthogonality#Orthonormality|orthonormal]]. * The [[LinearAlgebra/Diagonalization|diagonalization]] of a symmetric matrix is expressed as '''''A''' = '''QΛQ'''^-1^ = '''QΛQ'''^T^'' using the orthonormal eigenvectors. * The signs of the pivots are the same as the signs of the eigenvalues. [[LinearAlgebra/Projection|Projection matrices]] are always symmetric, and symmetric matrices are combinations of orthogonal projection matrices. Multiplying a rectangular matrix '''''R''''' by its transpose '''''R'''^T^'' will always create a symmetric matrix. This can be proven with the above properties: ''('''R'''^T^'''R''')^T^ = '''R'''^T^('''R'''^T^)^T^ = '''R'''^T^'''R'''''. |
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 square 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 is a square matrix of zeros with a one in each row. Multiplying a permutation matrix by some matrix A (PA) results in a row-exchanged A. Multiplying some matrix A by a permutation matrix (AP) results in a column-exhanged A.
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.
Symmetric Matrices
A symmetric matrix is equal to its transpose.
Only a square matrix can be symmetric.
These matrices have several useful properties:
The eigenvalues are always real.
The eigenvectors can be chosen to be orthonormal.
The diagonalization of a symmetric matrix is expressed as A = QΛQ-1 = QΛQT using the orthonormal eigenvectors.
- The signs of the pivots are the same as the signs of the eigenvalues.
Projection matrices are always symmetric, and symmetric matrices are combinations of orthogonal projection matrices.
Multiplying a rectangular matrix R by its transpose RT will always create a symmetric matrix. This can be proven with the above properties: (RTR)T = RT(RT)T = RTR.