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

A permutation matrix multiplied by matrix A returns matrix C which is a row-exchanged transformation of A.

For 3 by 3 matrices, there are 6 possible permutation matrices. They are often denoted based on the rows they exchange, such as P_{2 3}.

┌      ┐          ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐
│ 1 0 0│          │ 1 0 0│ │ 0 1 0│ │ 0 1 0│ │ 0 0 1│ │ 0 0 1│
│ 0 1 0│          │ 0 0 1│ │ 1 0 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│
│ 0 0 1│          │ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│ │ 1 0 0│
└      ┘          └      ┘ └      ┘ └      ┘ └      ┘ └      ┘
(identity matrix)   P        P        (and so on...)
                     2,3      1,2

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

CategoryRicottone