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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^''. | 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'''''. |
Matrix Transposition
For any matrix A, the transposition (AT) is a flipped version.
An alternative notation, found especially in matrix 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.
For orthogonal matrices (such as permutation matrices), the transpose is also the inverse: QT = Q-1. And because the left and right inverses are the same for any square matrix, QQT = QTQ.
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.