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



partial Derivatives of Eigen value decomposition or Singular value decomposition
To make Igor's more precise, Kato's book tells us that


1. 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 $M$.
2. 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.
3. 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.
4. 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.


