Special Matrices
Identity Matrix
The identity matrix multiplied by matrix A returns matrix A.
This matrix is simply a diagonal line of 1s in a matrix of 0s.
┌ ┐ │ 1 0 0│ │ 0 1 0│ │ 0 0 1│ └ ┘
Permutation Matrix
The permutation matrix multiplied by matrix A returns matrix C which is a mirrored transformation of A.
This matrix is simply a diagonal line of 1s in a matrix of 0s, but going the opposite direction as compared to an identity matrix
┌ ┐ │ 0 0 1│ │ 0 1 0│ │ 1 0 0│ └ ┘
Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order.
┌ ┐┌ ┐ ┌ ┐ │ 0 1││ 1 2│ │ 3 4│ │ 1 0││ 3 4│=│ 1 2│ └ ┘└ ┘ └ ┘ ┌ ┐┌ ┐ ┌ ┐ │ 1 2││ 0 1│ │ 2 1│ | 3 4|│ 1 0│=│ 4 3│ └ ┘└ ┘ └ ┘
Inverse Matrices
An inverse matrix is denoted as A-1. If a matrix is multiplied by its inverse matrix, it returns the identity matrix.
If A-1 exists, then A is invertible and non-singular. Not all matrices are invertible.
For a square matrix A, the left inverse is the same as the right inverse. AA-1 = A-1A = I