Handling Singular matrices in Algorithms

[beanz]

Hi All

I'm new to this forum so please be kind :)

I am doing a project on handling singular matrices in algorithms.

Basically what i have to do is to find out how to solve the system Ax=B when A is not square or det(A)=0. Because it does not have an inverse, I don't know what to do or how to approach this problem. Any help would be appreciated


P.S I tried the Guassian Elimination Method, but I have to find another way

 

[SpaceTiger]

P.S I tried the Guassian Elimination Method, but I have to find another way

What was wrong with gaussian elimination? I had a similar problem a while back (specifically, I needed to do it numerically) and I finally settled on writing a gaussian elimination routine.

 

[Hurkyl]

You could apply some sort of regularization. e.g. instead solve:

$$(A^T A + \lambda I) x = A^T b$$

(for your favorite λ > 0)

The system is no longer singular, but the answers you get will be biased away from the actual solutions. In particular, the answer will be unique, instead of getting a bunch of solutions.


I don't think this would help with Gaussian elimination, though. I share SpaceTiger's question -- what's wrong with it?

 

[beanz]

Thanx Guys,

I really appreciated your help. Nothings wrong with the gaussian elimination method. I just have to find out "various" ways of solving the system and tha Gaussian Elimination is just another method.

I have another question. Consider A , an n*n matrix which is invertible, and A^-1, its inverse. If A is transposed, how would it affect A^-1. (in other words how can i find the inverse of the Transpose of A without solving [A|I])

Thanx in advence

 

[LeBrad]

Look up Rank-Deficient Least Squares methods.