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| = Matrix Transposition = | = Transposition = '''Transposition''' is the process of 'flipping' a matrix. |
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| == Introduction == | == Description == |
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| The transpose of a matrix is a flipped version. The transpose of A is usually denoted A^T^; some notations, especially programming languages, instead use A'. | A transposed matrix is commonly notated with a ''T'' superscript, as in '''''A'''^T^''. In many programming languages however, the notation '''''A' ''''' is preferred. Cell (''i'',''j'') of '''''A'''^T^'' is equal to cell (''j'',''i'') of '''''A'''''. |
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| More formally, cell (''i'',''j'') of A^T^ is equal to cell (''j'',''i'') of A. ---- |
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| === Properties === | |
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| == Notable Properties == | The transpose of a product is the same as the reversed product of the transposed multiples. ''('''AB''')^T^ = '''B'''^T^ '''A'''^T^''. |
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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^. | [[LinearAlgebra/Invertibility|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''. |
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| Inversion and transposition can be done in any order: (A^-1^)^T^ = (A^T^)^-1^. | 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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| ---- | A [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric matrix]] is equal to its transpose: '''''A''' = '''A'''^T^''. Only square matrices can be symmetric. |
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== Symmetric Matrices == A '''symmetric matrix''' is is any matrix that is equal to its transpose. 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 |
Transposition does not change the [[LinearAlgebra/Determinant|determinant]] or the [[LinearAlgebra/Trace|trace]]: * ''|'''A'''| = |'''A'''^T^|'' * ''tr('''A''') = tr('''A'''^T^)'' |
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.
Transposition does not change the determinant or the trace:
|A| = |AT|
tr(A) = tr(AT)