Pseudo Inverse calculation

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.