most recent 30 from http://mathoverflow.net 2013-06-18T07:12:39Z http://mathoverflow.net/feeds/question/96297 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96297/eigenvalues-of-matrix-sum vkillion 2012-05-08T05:53:17Z 2012-05-08T05:53:17Z

<p>Hello,</p> <p>I have a linear algebra problem that I need help with.</p> <p>Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large Hermitian 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.</p> <p>H_constant - generated before the loop. Real and symmetric about the diagonal. Does not change after initial calculation.</p> <p>H_location - generated during each iteration. Diagonal.</p> <p>H_final - the addition of H_constant and H_location. Therefore, it is also real and symmetric about the diagonal.</p> <p>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.</p> <p>I hope I was able to explain the problem well enough. I hope someone is able to help me with this problem.</p> <p>Thank you,</p> <p>Vincent</p>