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

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

  1. Matrix Transposition
    1. Definition
      1. Properties
    2. Symmetry

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.

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.

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LinearAlgebra/MatrixTransposition (last edited 2024-01-27 21:22:51 by DominicRicottone)