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.