Exploiting a the inverse to determine eigen vectors?

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SUMMARY

This discussion focuses on the potential of using a matrix A and its inverse A^{-1} to efficiently calculate the eigenvalues and eigenvectors of A, particularly in the context of RCW(FMM) analysis of light on crossed gratings. The author identifies unique symmetries in a block matrix that complicate traditional eigenvalue calculations, noting that QR routines are ineffective while LU routines may offer a solution. The author proposes developing a method to leverage these symmetries for calculating the inverse, thereby facilitating the extraction of eigenvalues and eigenvectors.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors in linear algebra
  • Familiarity with block matrix structures and their properties
  • Knowledge of LU decomposition and QR factorization techniques
  • Basic principles of matrix inversion
NEXT STEPS
  • Research methods for exploiting block matrix symmetries in eigenvalue problems
  • Learn about LU decomposition techniques specific to block matrices
  • Investigate advanced matrix inversion algorithms that utilize symmetries
  • Explore the application of eigenvalue problems in RCW(FMM) analysis
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Researchers in applied mathematics, physicists working with light analysis, and mathematicians focusing on linear algebra and matrix theory will benefit from this discussion.

Jonnyb302
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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.
 
Physics news on Phys.org
The question for the eigenvalues of ##A## or of ##A^{-1}## is essentially the same, so you will not have additional useful information.
 

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