partial Derivatives of Eigen value decomposition or Singular value decomposition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:39:23Z http://mathoverflow.net/feeds/question/77057 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77057/partial-derivatives-of-eigen-value-decomposition-or-singular-value-decomposition partial Derivatives of Eigen value decomposition or Singular value decomposition Ashin 2011-10-03T16:51:27Z 2011-10-04T06:11:05Z <p>Hi All,</p> <p>Suppose I've a symmetric matrix $A_{N \times N} = (A_{ij})$ which has a eigen value decomposition $A = UDU'$. I would like to know under what conditions $\frac{\partial U}{\partial A_{ij}}$ exists for all $i,j = 1,2, \ldots, N$. I found the following paper which talks about estimating the Jacobian of the SVD transformation</p> <p><a href="http://www.ics.forth.gr/cvrl/publications/conferences/2000_eccv_SVD_jacobian.pdf" rel="nofollow">http://www.ics.forth.gr/cvrl/publications/conferences/2000_eccv_SVD_jacobian.pdf</a></p> <p>But its not very clear regarding the conditions that the matrix $A$ would need to satisfy. Any help is much appreciated.</p> <p>Thanks Ashin</p> http://mathoverflow.net/questions/77057/partial-derivatives-of-eigen-value-decomposition-or-singular-value-decomposition/77059#77059 Answer by Igor Rivin for partial Derivatives of Eigen value decomposition or Singular value decomposition Igor Rivin 2011-10-03T16:59:21Z 2011-10-03T16:59:21Z <p>Read Kato's peturbation theory for linear operators (chapter 2 is sufficient for your question), and all will be revealed.</p> http://mathoverflow.net/questions/77057/partial-derivatives-of-eigen-value-decomposition-or-singular-value-decomposition/77108#77108 Answer by Denis Serre for partial Derivatives of Eigen value decomposition or Singular value decomposition Denis Serre 2011-10-04T06:11:05Z 2011-10-04T06:11:05Z <p>To make Igor's more precise, Kato's book tells us that</p> <ol> <li>if an eigenvalue of a matrix $A$ is simple, then it extends as an analytic function $M\mapsto\lambda(M)$ defined in a neighbourhood of $A$, such that $\lambda(M)$ is an eigenvalue of $M$.</li> <li>if $s\mapsto A(s)$ is an analytic, one-parameter, family of real symmetric matrices, their eigenvalues $\lambda_1(s),\ldots,\lambda_n(s)$ can be arranged so that they are analytic functions. Mind that they are not in increasing order in general, because their order can change at values of $s$ for which $A(s)$ has a multiple eigenvalue.</li> <li>The previous result becomes false when the family depends upon several parameters. A typical example with two parameters is $$A(s,t)=\begin{pmatrix} s &amp; t \\ t &amp; -s \end{pmatrix},$$ for which the eigenvalues $\pm\sqrt{s^2+t^2}$ are even not $C^1$-functions.</li> <li>However, Weyl inequalities tell us that for real symmetric matrices, ${\rm dist}({\rm Sp}(B),{\rm Sp}(A))\le\|B-A\|$. Hence the eigenvalues are Lipschitz function, with unit Lipschitz constant.</li> </ol>