Pseudo inverse matrix helps to get a satisfactory solution to linear equation which don't have any exact solution. This technique is used for curve fitting .
The pseudo inverse can be calculated like this . If A is the matrix
The pseudo inverse is this : Inverse[Transpose[a] * A ] * Transpose[A]
Normal inverse exists only for square matrix. So we can call this pseudo inverse as a general matrix inverse method.
Consider the following linear equations
5X1 + 4 X2 = 9
3X1 + 2X2 = 5
X1 + x2 = 3
Actually this set of equations doesn't have any exact solution.
A X R
5 4 x1 9
3 2 * x2 = 5
1 1 3
Clearly we can't calculate the normal inverse of A because it is not square matrix.
The pseudoinverse of A is
-.5 1.5 -1.0
.833 -1.833 1.333
So [Pseudoinverse]* [R] give the values for X1 and X2, which is the nearst solution for the equations.