Real Matrices with Real Eigenvalue pre- and Post multiplied by a Diagonal Matrix












0














Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.






matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition







share|cite|improve this question



























    0














    Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.






    matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition







    share|cite|improve this question

























      0












      0








      0







      Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.






      matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition







      share|cite|improve this question













      Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.






      matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition




      matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 3 at 22:56









      Arthur

      47112




      47112






















          1 Answer
          1






          active

          oldest

          votes


















          1














          I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.



          Take
          $$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
          and
          $$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
          Then $DAD$ and $A + D$ has non-real eigenvalues.






          share|cite|improve this answer





















          • Thanks for the counterexample.
            – Arthur
            Jan 4 at 3:10











          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%2f3061119%2freal-matrices-with-real-eigenvalue-pre-and-post-multiplied-by-a-diagonal-matrix%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1














          I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.



          Take
          $$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
          and
          $$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
          Then $DAD$ and $A + D$ has non-real eigenvalues.






          share|cite|improve this answer





















          • Thanks for the counterexample.
            – Arthur
            Jan 4 at 3:10
















          1














          I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.



          Take
          $$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
          and
          $$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
          Then $DAD$ and $A + D$ has non-real eigenvalues.






          share|cite|improve this answer





















          • Thanks for the counterexample.
            – Arthur
            Jan 4 at 3:10














          1












          1








          1






          I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.



          Take
          $$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
          and
          $$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
          Then $DAD$ and $A + D$ has non-real eigenvalues.






          share|cite|improve this answer












          I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.



          Take
          $$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
          and
          $$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
          Then $DAD$ and $A + D$ has non-real eigenvalues.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 3 at 23:50









          Theo Bendit

          16.7k12148




          16.7k12148












          • Thanks for the counterexample.
            – Arthur
            Jan 4 at 3:10


















          • Thanks for the counterexample.
            – Arthur
            Jan 4 at 3:10
















          Thanks for the counterexample.
          – Arthur
          Jan 4 at 3:10




          Thanks for the counterexample.
          – Arthur
          Jan 4 at 3:10


















          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%2f3061119%2freal-matrices-with-real-eigenvalue-pre-and-post-multiplied-by-a-diagonal-matrix%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