Eigenvectors of decomposed matrix

[Cuthalion]

Hello everyone,

I've got the eigenvectors of a matrix H (Hessenberg matrix) obtained from the decomposition A=QHQ'.
Now I seek the eigenvectors of the matrix A. I've found somewhere that it should be : eigenvector_of_A=Q*eigenvector_of_H but some numerical test with MATLAB doesn't agree.
For the context: I'm using LAPACK (Fortran) routine to compute the eigenvectors of matrix for simulation purpose.

What's the solution ?

Thanks in advance,

David

 

[jvicens]

Hello David,

Thank you for sharing your question with us. It sounds like you are working with a Hessenberg matrix and are trying to find the eigenvectors of the original matrix A using the eigenvectors of H. While the formula you found (eigenvector_of_A=Q*eigenvector_of_H) seems like it should work, it's important to keep in mind that numerical tests may not always agree due to various factors such as rounding errors or different precision levels.

One possible solution is to use a different method for finding the eigenvectors of A. There are several techniques available, such as the power method, QR algorithm, or Jacobi method. Each method has its own advantages and disadvantages, so it may be helpful to try a few and see which one works best for your specific matrix.

Another option is to check the LAPACK routine you are using and make sure it is correctly computing the eigenvectors. You can also try using a different software, such as MATLAB's built-in eigenvector function, to compare the results and see if they agree.

I hope this helps and good luck with your simulation! If you have any further questions, please don't hesitate to reach out.