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.