The limit of a matrix norm divided by a power of its spectral radius












0












$begingroup$


Let $M$ be a quadratic matrix with spectral radius $lambda$.



It is known that $lambda=lim_{n to infty} (||M^n||)^{1/n}$.



I am now interested in the limit of $frac{||M^n||}{lambda^n}$ as $n$ goes to infinity. Is there any theorem that leads me from the known fact to the solution of this?



Thanks in advance!






linear-algebra functional-analysis linear-transformations







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Let $M$ be a quadratic matrix with spectral radius $lambda$.



    It is known that $lambda=lim_{n to infty} (||M^n||)^{1/n}$.



    I am now interested in the limit of $frac{||M^n||}{lambda^n}$ as $n$ goes to infinity. Is there any theorem that leads me from the known fact to the solution of this?



    Thanks in advance!






    linear-algebra functional-analysis linear-transformations







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $M$ be a quadratic matrix with spectral radius $lambda$.



      It is known that $lambda=lim_{n to infty} (||M^n||)^{1/n}$.



      I am now interested in the limit of $frac{||M^n||}{lambda^n}$ as $n$ goes to infinity. Is there any theorem that leads me from the known fact to the solution of this?



      Thanks in advance!






      linear-algebra functional-analysis linear-transformations







      share|cite|improve this question









      $endgroup$




      Let $M$ be a quadratic matrix with spectral radius $lambda$.



      It is known that $lambda=lim_{n to infty} (||M^n||)^{1/n}$.



      I am now interested in the limit of $frac{||M^n||}{lambda^n}$ as $n$ goes to infinity. Is there any theorem that leads me from the known fact to the solution of this?



      Thanks in advance!






      linear-algebra functional-analysis linear-transformations




      linear-algebra functional-analysis linear-transformations






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 30 at 20:40









      Micky MesserMicky Messer

      287




      287






















          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$

          There can be different cases. For example, if $M$ is the identity matrix, then the limit is $1$. If $M=begin{pmatrix}1&1\0&1end{pmatrix}$, then $lambda=1$, $M^n=begin{pmatrix}1&n\0&1end{pmatrix}$, and so the limit is $infty$. In general, the limit (if it exists at all) can be any number form $1$ to $+infty$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for the answer! Do you know of any condition that makes sure that the limit exists? For example in my case there exists an $N$ such that $M^N$ is positive.
            $endgroup$
            – Micky Messer
            Jan 31 at 17:26












          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%2f3094084%2fthe-limit-of-a-matrix-norm-divided-by-a-power-of-its-spectral-radius%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









          3












          $begingroup$

          There can be different cases. For example, if $M$ is the identity matrix, then the limit is $1$. If $M=begin{pmatrix}1&1\0&1end{pmatrix}$, then $lambda=1$, $M^n=begin{pmatrix}1&n\0&1end{pmatrix}$, and so the limit is $infty$. In general, the limit (if it exists at all) can be any number form $1$ to $+infty$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for the answer! Do you know of any condition that makes sure that the limit exists? For example in my case there exists an $N$ such that $M^N$ is positive.
            $endgroup$
            – Micky Messer
            Jan 31 at 17:26
















          3












          $begingroup$

          There can be different cases. For example, if $M$ is the identity matrix, then the limit is $1$. If $M=begin{pmatrix}1&1\0&1end{pmatrix}$, then $lambda=1$, $M^n=begin{pmatrix}1&n\0&1end{pmatrix}$, and so the limit is $infty$. In general, the limit (if it exists at all) can be any number form $1$ to $+infty$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for the answer! Do you know of any condition that makes sure that the limit exists? For example in my case there exists an $N$ such that $M^N$ is positive.
            $endgroup$
            – Micky Messer
            Jan 31 at 17:26














          3












          3








          3





          $begingroup$

          There can be different cases. For example, if $M$ is the identity matrix, then the limit is $1$. If $M=begin{pmatrix}1&1\0&1end{pmatrix}$, then $lambda=1$, $M^n=begin{pmatrix}1&n\0&1end{pmatrix}$, and so the limit is $infty$. In general, the limit (if it exists at all) can be any number form $1$ to $+infty$.






          share|cite|improve this answer









          $endgroup$



          There can be different cases. For example, if $M$ is the identity matrix, then the limit is $1$. If $M=begin{pmatrix}1&1\0&1end{pmatrix}$, then $lambda=1$, $M^n=begin{pmatrix}1&n\0&1end{pmatrix}$, and so the limit is $infty$. In general, the limit (if it exists at all) can be any number form $1$ to $+infty$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 30 at 20:44









          VladimirVladimir

          5,413618




          5,413618












          • $begingroup$
            Thanks for the answer! Do you know of any condition that makes sure that the limit exists? For example in my case there exists an $N$ such that $M^N$ is positive.
            $endgroup$
            – Micky Messer
            Jan 31 at 17:26


















          • $begingroup$
            Thanks for the answer! Do you know of any condition that makes sure that the limit exists? For example in my case there exists an $N$ such that $M^N$ is positive.
            $endgroup$
            – Micky Messer
            Jan 31 at 17:26
















          $begingroup$
          Thanks for the answer! Do you know of any condition that makes sure that the limit exists? For example in my case there exists an $N$ such that $M^N$ is positive.
          $endgroup$
          – Micky Messer
          Jan 31 at 17:26




          $begingroup$
          Thanks for the answer! Do you know of any condition that makes sure that the limit exists? For example in my case there exists an $N$ such that $M^N$ is positive.
          $endgroup$
          – Micky Messer
          Jan 31 at 17:26


















          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.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3094084%2fthe-limit-of-a-matrix-norm-divided-by-a-power-of-its-spectral-radius%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

          MongoDB - Not Authorized To Execute Command

          Image
          .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I have successfully enabled authorization on MongoDB and I have created an account on the admin database and then I created an account for my database called test. The following connection string to connect to my test database works successfully: mongo --host 192.168.17.52 --port 27017 -u user1 -p password --authenticationDatabase test Only problem I have now is, I cannot execute commands such as: show dbs. I get the following error when I try to do so: "errmsg" : "not authorized on admin to execute command { listDatabases: 1.0, lsid: { id: UUID("a1d5bc0d-bc58-485e-b232-270758a89455") }, $db: "admin...
          Read more

          Npm cannot find a required file even through it is in the searched directory

          Image
          up vote 0 down vote favorite I got that error: users-Air:pdfuploader user$ npm run start client@0.1.0 start /Users/user/Documents/pdfuploader react-scripts start Could not find a required file. Name: index.html Searched in: /Users/user/Documents/pdfuploader/public npm ERR! code ELIFECYCLE npm ERR! errno 1 npm ERR! client@0.1.0 start: `react-scripts start` npm ERR! Exit status 1 npm ERR! npm ERR! Failed at the client@0.1.0 start script. npm ERR! This is probably not a problem with npm. There is likely additional logging output above. npm ERR! A complete log of this run can be found in: npm ERR! /Users/user/.npm/_logs/2018-11-19T10_19_14_555Z-debug.log users-Air:pdfuploader user$ And the file is in /Users/user/Documents/pdfuploader/public as you can see in my github repository: https://github.c...
          Read more

          in spring boot 2.1 many test slices are not allowed anymore due to multiple @BootstrapWith

          Image
          1 I tried to upgrade a yummy sandwich made of two test slices (@JsonTest and @JdbcTest in my case, crunchy test code in between) adding spring boot 2.1 flavour to it. But it seems it was not much of a success. I cannot annotate my tests with many @...Test since they are now each bringing their own XxxTestContextBootstrapper. It used to work when they all used same SpringBootTestContextBootstrapper. @RunWith(SpringRunner.class) @JdbcTest @JsonTest public class Test { @Test public void test() { System.out.printn("Hello, World !"); } } The error I get from BootstrapUtils is illegalStateException : Configuration error: found multiple declarations of @BootstrapWith for test class I understand I might be doing something wrong here but is there an easy way I could load both Json and Jdbc contex...
          Read more