Matrix Transposition

For any matrix A, the transposition (A^T) is a flipped version.

An alternative notation, found especially in matrix programming languages like Stata, Julia, and MATLAB, is A' .

Definition

Cell (i,j) of A^T 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 = B^T A^T.

Inversion and transposition can be done in any order: (A^-1)^T = (A^T)^-1.

For orthogonal matrices (such as permutation matrices), the transpose is also the inverse: Q^T = Q^-1. And because the left and right inverses are the same for any square matrix, QQ^T = Q^TQ.

Symmetry

A symmetric matrix is equal to its transpose: A = A^T. 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^TR)^T = R^T(R^T)^T = R^TR.

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