Tuesday, March 3, 2009

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.



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