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For any matrix '''''A''''', the transposition ('''''A'''^T^'') is a flipped version. An alternative notation, found especially in programming languages like [[Stata]], [[Julia]], and [[MATLAB]], is '''''A' '''''. |
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| == Introduction == | == Definition == |
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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''''. | 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. | === Properties === 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^''. |
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| == Notable Properties == | == Symmetry == |
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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^. | 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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| Inversion and transposition can be done in any order: (A^-1^)^T^ = (A^T^)^-1^. ---- == 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 |
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 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.
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.