Is There a Matrix Equivalent to Green's Function for Differential Equations?

[daudaudaudau]

Matrix "Green's function"

Hi.

If you have a differential equation \mathcal L y=f where \mathcal L is some linear differential operator, then you can find a particular solution using the Green's function technique. It is then said that the Green's function is kind of the inverse to \mathcal L, even though \mathcal L might not really have an inverse. Is it possible to do something similar for matrices? I.e. if we have a matrix equation \mathbf Ax=b, is there some matrix that can give me a particular solution x=\mathbf Gb even though \mathbf A might not be invertible ?

 

[jasonRF]

Yes, there is something called a pseudoinverse:


http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse

Strang's "linear algebra and its applications" talks about this in an appendix, and I suspect a bunch of other books do as well.