Exploiting a the inverse to determine eigen vectors?

Jonnyb302

Hello, this is related to my research on RCW(FMM) analysis of light shone onto crossed gratings.

main question: can you exploit having matrix A and its inverse A^{-1} to calculate the eigen vectors/values of A in a more effective way?

backgroud:
so my problem is this: I have an unusual matrix with some strange symmetries and I need its eigen vectors. Only when I rewrite it in block form, do these symmetries become apparent. I would post it here but I am not at home so I do not have my saved LaTeX document with useful macros. But the symmetries have no real name, and most people will not be familiar with them.

I have searched quite a bit on the internet and can not find a way to exploit these block matrix symmetries. QR routines do not seem to go well with block matrices. While LU routines might.

However, I think I could devise a scheme to more efficiently calculate the inverse given these symmetries. Hence I am hopping to calculate the inverse, then exploit it to calculate the eigen values and vectors.