Matrix Inversion

Matrix inversion plays a major role in many multivariate techniques. While the inverse of a matrix is defined only for quadratic matrices, the concept of matrix inversion can also be extended to rectangular matrices by introducing the pseudo-inverse of a matrix.
 
 

Inverse Matrix

Given a square matrix A, the inverse matrix X is defined by the following equation: AX = IThe inverse of A is denoted by A-1 and is unique. Note that not all square matrices can be inverted. If an inverse for A exists, Ais called a regular or nonsingular matrix, otherwise A is a singular matrix. For regular matrices, the following equation is valid: AA-1 = A-1A = I.

The inverse of a matrix may be calculated using several algorithms, one of them being the Gauss Jordan algorithm.

Rules:

(A^-1) ^-1 = A

(A^-1) ^T = (A^T) ^-1

(AB)^-1 = B^-1A ^-1

(λA)^-1 = λ^-1A ^-1

Example:

since