Matrix Transposition
Introduction
The transpose of a matrix is a flipped version.
┌ ┐ ┌ ┐ │ 1 2│ │ 1 3│ │ 3 4│ -> │ 2 4│ └ ┘ └ ┘
The transpose of A is denoted AT.
More formally, cell (i,j) of AT 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 = BT AT.
Inversion and transposition can be done in any order: (A-1)T = (AT)-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 RT will always create a symmetric matrix. This can be proven with the above property:
(RT R)T = RT (RT)T = RT R