From: stewart@cs.umd.edu (G. W. Stewart)
Newsgroups: sci.math.num-analysis
Subject: Re: Eigenvectors of the sum of two symmetric matrices
Date: 8 Jan 93 07:07:58 GMT
Organization: U of Maryland, Dept. of Computer Science, Coll. Pk., MD 20742

In article <1993Jan8.001959.13695@ccu1.aukuni.ac.nz> mmt@ccu1.aukuni.ac.nz (Mark Thomson) writes:
>If and A and B are two symmetric matrices, can anyone tell me whether it
>is possible to relate the eigenvectors of (A+B) and their corresponding
>eigenvalues to the eigenvectors and eigenvalues of A and B.
>
>Any help appreciated.
>
>Mark Thomson
>University of Auckland.

In general one can not say very much. If $\alpha_i$, $\beta_i$ and
$\gamma_i$ are the eigenvalues of $A$, $B$, and $C=A+B$, arranged
in ascending order, then
\[
\alpha_i+\beta_1 \leq \gamma_i \leq \alpha_i + \beta_n,
\]
where $n$ is the order of the matrices in question.  More results
of this nature can be found in my book with Sun, *Matrix Perturbation
Theory*.  Also in Horn and Johnson's excellent books *Matrix Analysis*
and *Topics in Matrix Analysis*.

More can be said if $B$ is of low rank.  The subject is to big to
go into here, and instead I am appending a bibliography.

Pete Stewart

\begin{thebibliography}{1}

\bibitem{argg:88}
P.~Arbenz, W.~Gander, and G.~H. Golub.
\newblock Restricted rank modification of the symmetric eigenvalue problem:
  Theoretical considerations.
\newblock {\em Linear Algebra and Its Applications}, 104:75--96, 1988.

\bibitem{argo:88}
P.~Arbenz and G.~H. Golub.
\newblock On the spectral decomposition of {H}ermitian matrices subject to
  indefinite low rank perturbations with applications.
\newblock {\em {SIAM} Journal on Matrix Analysis and Applications}, 9:40--58,
  1988.
\newblock Cited in \cite{govl:89}.

\bibitem{buns:78}
J.~R. Bunch, C.~P. Nielsen, and D.~C. Sorensen.
\newblock Rank one modification of the symmetric eigenproblem.
\newblock {\em Numerische Mathematik}, 31:31--48, 1978.

\bibitem{cupp:81}
J.~J.~M. Cuppen.
\newblock A divide and conquer method for the symmetric eigenproblem.
\newblock {\em Numerische Mathematik}, 36:177--195, 1981.

\bibitem{doso:87}
J.~J. Dongarra and D.~C. Sorensen.
\newblock A fully parallel algorithm for the symmetric eigenvalue problem.
\newblock {\em SIAM Journal on Scientific and Statistical Computing},
  8:s139--s154, 1987.

\bibitem{govl:89}
G.~H. Golub and C.~F. Van~Loan.
\newblock {\em Matrix Computations}.
\newblock Johns Hopkins University Press, Baltimore, Maryland, 2nd edition,
  1989.

\bibitem{guei:92}
M.~Gu and S.~C. Eisenstat.
\newblock A stable and efficient algorithm for the rank-one modification of the
  symmetric eigenvalue problem.
\newblock Research Report YALEU/DCS/RR-916, Department of Computer Science,
  Yale University, 1992.

\bibitem{movv:91a}
M.~Moonen, P.~Van~Dooren, and J.~Vandewalle.
\newblock A note on ``{E}fficient numerically stabilized rank-one
  eigenstructure updating.
\newblock {\em IEEE Transactions on Signal Processing}, 39:1911--1913, 1991.
\newblock Reply by DeGroat, pp.~1913--1914.

\bibitem{sota:91}
D.~C. Sorensen and P.~T.~P. Tang.
\newblock On the orthogonality of eigenvectors computed by divide and conquer
  techniques.
\newblock {\em SIAM Journal on Numerical Analysis}, 28:1752--1775, 1990.

\end{thebibliography}