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Math Help - Eigenvalues of Matrix Addition

  1. #1
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    Eigenvalues of Matrix Addition

    Hello,

    I have a linear algebra problem that I need help with.

    Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large matrices (6^n x 6^n, where n>= 3, to be specific). Currently, we are just using MATLAB's eig() function to get them. I am trying to find optimizations for the simulations to cut down on computing time. There are three matrices that we use.

    H_constant - generated before the loop. Real and symmetric about the diagonal. Does not change after initial calculation.

    H_location - generated during each iteration. Diagonal.

    H_final - H_constant + H_location. Therefore, it is also real and symmetric about the diagonal.

    It is H_final that we need the eigenvalues and eigenvectors of. My theory is that we calculate the eigenvalues and eigenvectors of H_constant (which won't change after the initial calculation) once. We use this result with the eigenvalues of H_location (the diagonal), to get the eigenvalues and eigenvectors of H_final1. This would reduce our computation from tens of thousands of eig() calls to 1 eig() call and tens of thousands of very simple operations. I don't remember enough of my linear algebra to prove such a theory.

    I hope I was able to explain the problem well enough. I hope someone is able to help me with this problem.

    Thank you,

    Vincent
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  2. #2
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    Re: Eigenvalues of Matrix Addition

    Do you mean that H_final = H_constant + H_location?

    Are all matrices nonsingular?
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  3. #3
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    Re: Eigenvalues of Matrix Addition

    Thank you for your response.

    Yes. H_final = H_constant + H_location and the matrices are nonsingular.
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