Matrix Transposition

Introduction

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'.

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

More formally, cell (i,j) of A^T 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 = B^T A^T.

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

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