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| = MatrixTransposition = | = Matrix Transposition = <<TableOfContents>> ----- |
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| The transpose of a matrix is a flipped version. | 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'. |
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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^. | More formally, cell (''i'',''j'') of A^T^ is equal to cell (''j'',''i'') of A. ---- |
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| == Multiplication of Transposed Matrices == | == Notable Properties == |
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| Inversion and transposition can be done in any order: (A^-1^)^T^ = (A^T^)^-1^. ---- |
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| == Inverses of Transposed Matrices == | |
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| A simple proof based on the definition of inverse matrices and the above multiplicative property: | == Symmetric Matrices == |
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| {{{ -1 -1 A A = I = A A |
A '''symmetric matrix''' is is any matrix that is equal to its transpose. |
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| (leave the left side off for now) | 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: |
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| -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. |
(R^T^ R)^T^ = R^T^ (R^T^)^T^ = R^T^ R |
Matrix Transposition
Introduction
The transpose of a matrix is a flipped version. The transpose of A is usually denoted AT; some notations, especially programming languages, instead use 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 4More formally, cell (i,j) of AT is equal to cell (j,i) of A.
Notable Properties
The transpose of a product is the same as the reversed product of the transposed multiples. (A B)T = BT AT.
Inversion and transposition can be done in any order: (A-1)T = (AT)-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 RT will always create a symmetric matrix. This can be proven with the above property:
(RT R)T = RT (RT)T = RT R