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| ## page was renamed from LinearAlgebra/MatrixTransposition = Matrix Transposition = |
= Transposition = |
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| 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' '''''. |
'''Transposition''' is the process of 'flipping' a matrix. |
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| == Definition == | == Description == A transposed matrix is commonly notated with a ''T'' superscript, as in '''''A'''^T^''. In many programming languages however, the notation '''''A' ''''' is preferred. |
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| [[LinearAlgebra/MatrixInversion|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''. | [[LinearAlgebra/Invertibility|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''. |
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| 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'''''. | For [[LinearAlgebra/Orthogonality|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the transpose and inverse are equivalent: '''''Q'''^T^ = '''Q'''^-1^''. |
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| ---- == Symmetry == A [[LinearAlgebra/MatrixProperties#Symmetry|symmetric]] matrix is equal to its transpose: '''''A''' = '''A'''^T^''. Only square matrices (''n'' by ''n'') can be symmetric. 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'''''. |
A [[LinearAlgebra/SpecialMatrices#SymmetricMatrices|symmetric matrix]] is equal to its transpose: '''''A''' = '''A'''^T^''. Only square matrices can be symmetric. |
Transposition
Transposition is the process of 'flipping' a matrix.
Contents
Description
A transposed matrix is commonly notated with a T superscript, as in AT. In many programming languages however, the notation A' is preferred.
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 and inverse are equivalent: QT = Q-1.
A symmetric matrix is equal to its transpose: A = AT. Only square matrices can be symmetric.