partial Derivatives of Eigen value decomposition or Singular value decomposition - MathOverflow

partial Derivatives of Eigen value decomposition or Singular value decomposition

2011-10-03T16:51:27Z

Hi All,

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

http://www.ics.forth.gr/cvrl/publications/conferences/2000_eccv_SVD_jacobian.pdf

But its not very clear regarding the conditions that the matrix $A$ would need to satisfy. Any help is much appreciated.

Thanks Ashin

Answer by Igor Rivin for partial Derivatives of Eigen value decomposition or Singular value decomposition

2011-10-03T16:59:21Z

Read Kato's peturbation theory for linear operators (chapter 2 is sufficient for your question), and all will be revealed.

Answer by Denis Serre for partial Derivatives of Eigen value decomposition or Singular value decomposition

2011-10-04T06:11:05Z

To make Igor's more precise, Kato's book tells us that

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 $M$.
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.
The previous result becomes false when the family depends upon several parameters. A typical example with two parameters is $$A(s,t)=\begin{pmatrix} s & t \\ t & -s \end{pmatrix},$$ for which the eigenvalues $\pm\sqrt{s^2+t^2}$ are even not $C^1$-functions.
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.