Perturbation theory of the eigenvalues about a symmetric matrix: Reality of Eigenvalues












0














Let $A$ be an $ntimes n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $hat A := A+E$ are all real? What if we have more properties on $A$ and $E$: $A$ is diagonalizable and $E$ is skew symmetric?



Note:
I have seen two interesting lines of discussions on the eigenvalues of a symmetric matrix with perturbation here and here, but there is no discussion on the reality of eigenvalues.
Thanks.






linear-algebra matrices eigenvalues-eigenvectors matrix-decomposition perturbation-theory







share|cite|improve this question


















  • 2




    Have you lookes at examples like :$$A= begin{bmatrix} 0 & 0 \ 0 & 0 \ end{bmatrix} $$and $$E= begin{bmatrix} 0 & epsilon \ - epsilon & 0 \ end{bmatrix}. $$
    – Keith McClary
    Jan 4 at 22:35






  • 1




    Going beyond Keith's excellent example, it doesn't matter what $A$ is, there are always going to be matrices arbitrarily close to $A$ with non-real eigenvalues. This is pretty much because the reals have no interior as a subset of the complex numbers, so all real numbers are approached by non-reals.
    – Paul Sinclair
    Jan 5 at 3:50










  • Thank you very much for the excellent example and the excellent intuition!
    – Arthur
    Jan 5 at 5:38
















0














Let $A$ be an $ntimes n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $hat A := A+E$ are all real? What if we have more properties on $A$ and $E$: $A$ is diagonalizable and $E$ is skew symmetric?



Note:
I have seen two interesting lines of discussions on the eigenvalues of a symmetric matrix with perturbation here and here, but there is no discussion on the reality of eigenvalues.
Thanks.






linear-algebra matrices eigenvalues-eigenvectors matrix-decomposition perturbation-theory







share|cite|improve this question


















  • 2




    Have you lookes at examples like :$$A= begin{bmatrix} 0 & 0 \ 0 & 0 \ end{bmatrix} $$and $$E= begin{bmatrix} 0 & epsilon \ - epsilon & 0 \ end{bmatrix}. $$
    – Keith McClary
    Jan 4 at 22:35






  • 1




    Going beyond Keith's excellent example, it doesn't matter what $A$ is, there are always going to be matrices arbitrarily close to $A$ with non-real eigenvalues. This is pretty much because the reals have no interior as a subset of the complex numbers, so all real numbers are approached by non-reals.
    – Paul Sinclair
    Jan 5 at 3:50










  • Thank you very much for the excellent example and the excellent intuition!
    – Arthur
    Jan 5 at 5:38














0












0








0







Let $A$ be an $ntimes n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $hat A := A+E$ are all real? What if we have more properties on $A$ and $E$: $A$ is diagonalizable and $E$ is skew symmetric?



Note:
I have seen two interesting lines of discussions on the eigenvalues of a symmetric matrix with perturbation here and here, but there is no discussion on the reality of eigenvalues.
Thanks.






linear-algebra matrices eigenvalues-eigenvectors matrix-decomposition perturbation-theory







share|cite|improve this question













Let $A$ be an $ntimes n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $hat A := A+E$ are all real? What if we have more properties on $A$ and $E$: $A$ is diagonalizable and $E$ is skew symmetric?



Note:
I have seen two interesting lines of discussions on the eigenvalues of a symmetric matrix with perturbation here and here, but there is no discussion on the reality of eigenvalues.
Thanks.






linear-algebra matrices eigenvalues-eigenvectors matrix-decomposition perturbation-theory




linear-algebra matrices eigenvalues-eigenvectors matrix-decomposition perturbation-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 4 at 17:12









ArthurArthur

47112




47112








  • 2




    Have you lookes at examples like :$$A= begin{bmatrix} 0 & 0 \ 0 & 0 \ end{bmatrix} $$and $$E= begin{bmatrix} 0 & epsilon \ - epsilon & 0 \ end{bmatrix}. $$
    – Keith McClary
    Jan 4 at 22:35






  • 1




    Going beyond Keith's excellent example, it doesn't matter what $A$ is, there are always going to be matrices arbitrarily close to $A$ with non-real eigenvalues. This is pretty much because the reals have no interior as a subset of the complex numbers, so all real numbers are approached by non-reals.
    – Paul Sinclair
    Jan 5 at 3:50










  • Thank you very much for the excellent example and the excellent intuition!
    – Arthur
    Jan 5 at 5:38














  • 2




    Have you lookes at examples like :$$A= begin{bmatrix} 0 & 0 \ 0 & 0 \ end{bmatrix} $$and $$E= begin{bmatrix} 0 & epsilon \ - epsilon & 0 \ end{bmatrix}. $$
    – Keith McClary
    Jan 4 at 22:35






  • 1




    Going beyond Keith's excellent example, it doesn't matter what $A$ is, there are always going to be matrices arbitrarily close to $A$ with non-real eigenvalues. This is pretty much because the reals have no interior as a subset of the complex numbers, so all real numbers are approached by non-reals.
    – Paul Sinclair
    Jan 5 at 3:50










  • Thank you very much for the excellent example and the excellent intuition!
    – Arthur
    Jan 5 at 5:38








2




2




Have you lookes at examples like :$$A= begin{bmatrix} 0 & 0 \ 0 & 0 \ end{bmatrix} $$and $$E= begin{bmatrix} 0 & epsilon \ - epsilon & 0 \ end{bmatrix}. $$
– Keith McClary
Jan 4 at 22:35




Have you lookes at examples like :$$A= begin{bmatrix} 0 & 0 \ 0 & 0 \ end{bmatrix} $$and $$E= begin{bmatrix} 0 & epsilon \ - epsilon & 0 \ end{bmatrix}. $$
– Keith McClary
Jan 4 at 22:35




1




1




Going beyond Keith's excellent example, it doesn't matter what $A$ is, there are always going to be matrices arbitrarily close to $A$ with non-real eigenvalues. This is pretty much because the reals have no interior as a subset of the complex numbers, so all real numbers are approached by non-reals.
– Paul Sinclair
Jan 5 at 3:50




Going beyond Keith's excellent example, it doesn't matter what $A$ is, there are always going to be matrices arbitrarily close to $A$ with non-real eigenvalues. This is pretty much because the reals have no interior as a subset of the complex numbers, so all real numbers are approached by non-reals.
– Paul Sinclair
Jan 5 at 3:50












Thank you very much for the excellent example and the excellent intuition!
– Arthur
Jan 5 at 5:38




Thank you very much for the excellent example and the excellent intuition!
– Arthur
Jan 5 at 5:38










0






active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061844%2fperturbation-theory-of-the-eigenvalues-about-a-symmetric-matrix-reality-of-eige%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061844%2fperturbation-theory-of-the-eigenvalues-about-a-symmetric-matrix-reality-of-eige%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

An IMO inspired problem

Image
14 4 This problem from IMO 1988 is said to be one of the most elegant ones in functional equations. Problem : The function $f$ is defined on the set of all positive integers as follows: begin{align} f(1) = 1, f(3) &= 3, f(2n) = f(n), \ f(4n+1) &= 2f(2n+1) - f(n), \ f(4n+3) &= 3 f(2n+1) - 2f(n) end{align} Find the number of $n$ with $f(n) = n, 1 leq n leq 1988$. The main idea towards the solution is realizing that these conditions stand for the fact that $f(n)$ just reverses the digits in the binary representation of the number $n$. So, essentially the solution is finding the number of binary pallindromes $leq 1988_{10}$. My question is the following : How to reformulate the problem so that $f(n)$ reverses the digits of $n$ in its ternary representation. Or even better, can we reformulate it for a
Read more

Management

Image
Management From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Manager" redirects here. For other uses, see Management (disambiguation) and Manager (disambiguation). An organization chart for the United States Coast Guard shows the hierarchy of managerial roles in that organization. Business administration Management of a business Accounting Management accounting Financial accounting Financial audit Business entities Cooperative Corporation Limited liability company Partnership Sole proprietorship State-owned enterprise Corporate governance Annual general meeting Board of directors Supervisory board Advisory board Audit committee Corporate law Commercial law Constitutional documents Contract Corporate crime Corporate liability Insolvency law International trade law Mergers and acquisitions Economics Commodity Pu
Read more

Has there ever been an instance of an active nuclear power plant within or near a war zone?

Image
36 3 The question Has there ever been a military combat going on around an active (running or in outage but not decommissioned) commercial nuclear power plant? Or dangerously near it? If yes, how did the plant and its personnel fare in such situation? Why I ask Let's assume for the scope of this question modern commercial nuclear power plants (CANDU, PWR, BWR of generation II plus all of gen III and newer) are usually reasonably safe to operate given that the engineering has been done right and there is a whole cohort of very well trained personnel on-site at all times, well rested and with considerable resources on their hands. Things hard to get with an armed conflict like the recent Syrian war raging around. I can't seem to find any reference or a comprehensive article about this, hence this question.
Read more