Sum of Matrices Eigenvalues

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

Answer 1

[John D'Errico]

Sorry, but it looks like your theory needs some work. Knowing the eigenvalues and eigenvectors of one matrix will not tell you anything about those parameters for a different matrix in general, even with so simple a modification. Only if that H_location matrix is a constant times an identity matrix will your problem have a simple solution.

I.e., you need for the diagonal matrix to be constant down that main diagonal. Then we can simply add that constant to the eigenvalues of your original matrix, with no change to the eigenvectors. This works because any set of orthogonal basis vectors will form a valid set of eigenvectors for an identity matrix.

Just wanting a problem to be simple does not always make it so.