most recent 30 from http://mathoverflow.net 2013-05-20T18:00:46Z http://mathoverflow.net/feeds/question/93337 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93337/bandwidth-reduction-of-multiple-matrices Greg von Winckel 2012-04-06T16:35:06Z 2012-06-16T13:22:00Z

<p>Suppose I have a symmetric matrix A, and several diagonal matrices $D_1,D_2,...$ </p> <p>Are there any matrix transformations, such as $P^\top A P$ so that $P^\top AP$, $P^\top D_1 P$, $P^\top D_2 P$, etc are either all tridiagonal, or all have minimimal bandwidth in some sense? If for example, I only had $D_1$ then solving the generalized eigenvalue problem for the matrix pencil $A,D_1$ would give me a basis that simultaneously diagonalizes both $A$ and $D_1$. Obviously, one basis will not simultaneously diagonalize more matrices in general, but a set of banded matrices would also be pretty nice. </p> <p>I am aware of Tisseur and Garvey's papers on simultaneous tridiagonalization of two matrices. </p>

http://mathoverflow.net/questions/93337/bandwidth-reduction-of-multiple-matrices/93395#93395 Bart 2012-04-07T08:25:00Z 2012-04-07T08:25:00Z

<p>If you use the typical reorderings (like reverse Cuthill-McKee ordering), $P^T A P$ will have a smaller bandwidth (in general, not tridiagonal though). Since $P$ is a permutation matrix, all $P^T D_i P$ will remain diagonal too.</p> <p>Bart</p>