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| = Matrix Transposition = | = Transposition = '''Transposition''' is the process of 'flipping' a matrix. <<TableOfContents>> ----- |
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| == Introduction == | == Description == |
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| The transpose of a matrix is a flipped version. | 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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| ┌ ┐ ┌ ┐ │ 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^''. |
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| [[LinearAlgebra/Invertibility|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''. | |
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| 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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| == Inverses of Transposed Matrices == | 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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| A simple proof based on the definition of inverse matrices and the above multiplicative property: {{{ -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. |
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)