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| = MatrixTransposition = | = Matrix Transposition = For any matrix '''''A''''', the transposition ('''''A'''^T^'') is a flipped version. An alternative notation, found especially in matrix programming languages like [[Stata]], [[Julia]], and [[MATLAB]], is '''''A' '''''. <<TableOfContents>> ----- |
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| == Introduction == | == Definition == |
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| The transpose of a matrix is a flipped version. | Cell (''i'',''j'') of '''''A'''^T^'' is equal to cell (''j'',''i'') of '''''A'''''. |
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| ┌ ┐ ┌ ┐ │ 1 2│ │ 1 3│ │ 3 4│ -> │ 2 4│ └ ┘ └ ┘ |
julia> A = [1 2; 3 4] 2×2 Matrix{Int64}: 1 2 3 4 julia> A' 2×2 adjoint(::Matrix{Int64}) with eltype Int64: 1 3 2 4 |
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The transpose of A is denoted A^T^. |
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| == Multiplication of Transposed Matrices == | === Properties === |
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| The transpose of a product is the same as the reversed product of the transposed multiples. (A B)^T^ = B^T^ A^T^. | The transpose of a product is the same as the reversed product of the transposed multiples. ''('''AB''')^T^ = '''B'''^T^ '''A'''^T^''. [[LinearAlgebra/MatrixInversion|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''. For [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the transpose is also the [[LinearAlgebra/MatrixInversion|inverse]]: '''''Q'''^T^ = '''Q'''^-1^''. And because the left and right inverses are the same for any square matrix, '''''QQ'''^T^ = '''Q'''^T^'''Q'''''. ---- |
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| == Inverses of Transposed Matrices == | == Symmetry == |
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| A simple proof based on the definition of inverse matrices and the above multiplicative property: | A [[LinearAlgebra/MatrixProperties#Symmetry|symmetric]] matrix is equal to its transpose: '''''A''' = '''A'''^T^''. Only square matrices (''n'' by ''n'') can be symmetric. |
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| {{{ -1 -1 A A = I = A A (leave the left side off for now) -1 A A = I T -1 T A A = I T -1 A A = I T -1 T A A = I (bring back the left side) T -1 T -1 A A = I = A A (and it should now be clear that) T -1 -1 T A = A }}} Inverses and transposes can be done in any order. |
However, multiplying a rectangular matrix '''''R''''' by its transpose '''''R'''^T^'' will always create a symmetric matrix. This can be proven with the above property: ''('''R'''^T^'''R''')^T^ = '''R'''^T^('''R'''^T^)^T^ = '''R'''^T^'''R'''''. |
Matrix Transposition
For any matrix A, the transposition (AT) is a flipped version.
An alternative notation, found especially in matrix programming languages like Stata, Julia, and MATLAB, is A' .
Contents
Definition
Cell (i,j) of AT is equal to cell (j,i) of A.
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> A'
2×2 adjoint(::Matrix{Int64}) with eltype Int64:
1 3
2 4
Properties
The transpose of a product is the same as the reversed product of the transposed multiples. (AB)T = BT AT.
Inversion and transposition can be done in any order: (A-1)T = (AT)-1.
For orthogonal matrices (such as permutation matrices), the transpose is also the inverse: QT = Q-1. And because the left and right inverses are the same for any square matrix, QQT = QTQ.
Symmetry
A symmetric matrix is equal to its transpose: A = AT. Only square matrices (n by n) can be symmetric.
However, multiplying a rectangular matrix R by its transpose RT will always create a symmetric matrix. This can be proven with the above property: (RTR)T = RT(RT)T = RTR.