How much can a diagonal matrix change the eigenvalues of a symmetric matrix? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:36:15Z http://mathoverflow.net/feeds/question/76015 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76015/how-much-can-a-diagonal-matrix-change-the-eigenvalues-of-a-symmetric-matrix How much can a diagonal matrix change the eigenvalues of a symmetric matrix? Anadim 2011-09-21T01:06:08Z 2011-09-23T09:59:37Z <p>Suppose that we have a symmetric matrix ${\bf S}$ with eigenvalue decomposition ${\bf S} = {\bf Q}{\bf \Lambda}{\bf Q}^T$. Assume that we have two diagonal matrices ${\bf D}_1$ and ${\bf D}_2$ that are multiplying ${\bf S}$ from the left and right, i.e. ${\bf A} = {\bf D}_1{\bf S}{\bf D}_2$. Can we relate the eigenvalues of ${\bf S}$ to the ones of ${\bf A}$? How about the case where ${\bf S}$ is not symmetric?</p> http://mathoverflow.net/questions/76015/how-much-can-a-diagonal-matrix-change-the-eigenvalues-of-a-symmetric-matrix/76197#76197 Answer by A. Lerario for How much can a diagonal matrix change the eigenvalues of a symmetric matrix? A. Lerario 2011-09-23T09:59:37Z 2011-09-23T09:59:37Z <p>In general there is no relation: for example consider the simplest case $S$ itself is diagonal and invertible. Letting $D_1=S^{-1}$ then $A$ can be any diagonal matrix $D_2$. The only considerations you can do are related to the presence of the zero eigenvalues using Binet formula for determinants. Notice also that in general $A$ itself can be nonsymmetric, and its eigenvalues can be complex. However small perturbations, i.e. small $D_1$ and $D_2$, result in a small perturbation of the eigenvalues of $S$ in the complex plane.</p>