What does the following notation mean: $(L-lambda I)$












1












$begingroup$


In my tekstbook powervectors are discussed where
$(L-lambda I)^p textbf{v}=0$ was mentioned. Here is $L$ a linear operator on the vectorspace $V$, $lambda$ a scalar and $textbf{v} in V $.



My question: How is $L-lambda I$ definied? I'm confused because $L$ is here a function and not a matrix.






linear-algebra notation definition







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You can think " I " as an identity function not a matrix
    $endgroup$
    – Euduardo
    Dec 10 '18 at 14:31
















1












$begingroup$


In my tekstbook powervectors are discussed where
$(L-lambda I)^p textbf{v}=0$ was mentioned. Here is $L$ a linear operator on the vectorspace $V$, $lambda$ a scalar and $textbf{v} in V $.



My question: How is $L-lambda I$ definied? I'm confused because $L$ is here a function and not a matrix.






linear-algebra notation definition







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You can think " I " as an identity function not a matrix
    $endgroup$
    – Euduardo
    Dec 10 '18 at 14:31














1












1








1





$begingroup$


In my tekstbook powervectors are discussed where
$(L-lambda I)^p textbf{v}=0$ was mentioned. Here is $L$ a linear operator on the vectorspace $V$, $lambda$ a scalar and $textbf{v} in V $.



My question: How is $L-lambda I$ definied? I'm confused because $L$ is here a function and not a matrix.






linear-algebra notation definition







share|cite|improve this question











$endgroup$




In my tekstbook powervectors are discussed where
$(L-lambda I)^p textbf{v}=0$ was mentioned. Here is $L$ a linear operator on the vectorspace $V$, $lambda$ a scalar and $textbf{v} in V $.



My question: How is $L-lambda I$ definied? I'm confused because $L$ is here a function and not a matrix.






linear-algebra notation definition




linear-algebra notation definition






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 10 '18 at 16:10







Keep_On_Cruising

















asked Dec 10 '18 at 14:29









Keep_On_CruisingKeep_On_Cruising

1719




1719








  • 1




    $begingroup$
    You can think " I " as an identity function not a matrix
    $endgroup$
    – Euduardo
    Dec 10 '18 at 14:31














  • 1




    $begingroup$
    You can think " I " as an identity function not a matrix
    $endgroup$
    – Euduardo
    Dec 10 '18 at 14:31








1




1




$begingroup$
You can think " I " as an identity function not a matrix
$endgroup$
– Euduardo
Dec 10 '18 at 14:31




$begingroup$
You can think " I " as an identity function not a matrix
$endgroup$
– Euduardo
Dec 10 '18 at 14:31










2 Answers
2






active

oldest

votes


















7












$begingroup$

$I$ is the identity operator on the vector space: $Ix = x$ for all $x in V$.
So $$(L - lambda I) x = L x - lambda I x = L x - lambda x$$






share|cite|improve this answer









$endgroup$





















    3












    $begingroup$

    Robert did fully answer the question, but I'd like to add a different example: The differential equation of the harmonic oscillator is
    $$frac{mathrm{d}^2 u(t)}{mathrm{d}t^2}+2beta frac{mathrm{d}u(t)}{mathrm{d}t}+omega^2 u(t)=0$$
    And it can be rewritten as
    $$(D^2+2beta D+omega^2 I)u(t)=0$$
    Where $D$ is the differential operator: $Du(t)=frac{mathrm{d}u(t)}{mathrm{d}t}$.

    Or the Klein-Gordon equation in $1+1$ dimensions:
    $$left(frac{1}{c^2} partial_t^2-partial_x^2+mu Iright)u(t)=0$$
    This kind of notation is very useful, because the differential operator will take it's eigenvalue on it's eigenfunction, so you can easily get the dispersion relation, with just substitutions of the eigenvalues: $partial_t to (-i omega)$ and $partial_x to (ik)$ in the case of a plane wave.






    share|cite|improve this answer











    $endgroup$













      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%2f3033983%2fwhat-does-the-following-notation-mean-l-lambda-i%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      7












      $begingroup$

      $I$ is the identity operator on the vector space: $Ix = x$ for all $x in V$.
      So $$(L - lambda I) x = L x - lambda I x = L x - lambda x$$






      share|cite|improve this answer









      $endgroup$


















        7












        $begingroup$

        $I$ is the identity operator on the vector space: $Ix = x$ for all $x in V$.
        So $$(L - lambda I) x = L x - lambda I x = L x - lambda x$$






        share|cite|improve this answer









        $endgroup$
















          7












          7








          7





          $begingroup$

          $I$ is the identity operator on the vector space: $Ix = x$ for all $x in V$.
          So $$(L - lambda I) x = L x - lambda I x = L x - lambda x$$






          share|cite|improve this answer









          $endgroup$



          $I$ is the identity operator on the vector space: $Ix = x$ for all $x in V$.
          So $$(L - lambda I) x = L x - lambda I x = L x - lambda x$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 10 '18 at 14:31









          Robert IsraelRobert Israel

          322k23212465




          322k23212465























              3












              $begingroup$

              Robert did fully answer the question, but I'd like to add a different example: The differential equation of the harmonic oscillator is
              $$frac{mathrm{d}^2 u(t)}{mathrm{d}t^2}+2beta frac{mathrm{d}u(t)}{mathrm{d}t}+omega^2 u(t)=0$$
              And it can be rewritten as
              $$(D^2+2beta D+omega^2 I)u(t)=0$$
              Where $D$ is the differential operator: $Du(t)=frac{mathrm{d}u(t)}{mathrm{d}t}$.

              Or the Klein-Gordon equation in $1+1$ dimensions:
              $$left(frac{1}{c^2} partial_t^2-partial_x^2+mu Iright)u(t)=0$$
              This kind of notation is very useful, because the differential operator will take it's eigenvalue on it's eigenfunction, so you can easily get the dispersion relation, with just substitutions of the eigenvalues: $partial_t to (-i omega)$ and $partial_x to (ik)$ in the case of a plane wave.






              share|cite|improve this answer











              $endgroup$


















                3












                $begingroup$

                Robert did fully answer the question, but I'd like to add a different example: The differential equation of the harmonic oscillator is
                $$frac{mathrm{d}^2 u(t)}{mathrm{d}t^2}+2beta frac{mathrm{d}u(t)}{mathrm{d}t}+omega^2 u(t)=0$$
                And it can be rewritten as
                $$(D^2+2beta D+omega^2 I)u(t)=0$$
                Where $D$ is the differential operator: $Du(t)=frac{mathrm{d}u(t)}{mathrm{d}t}$.

                Or the Klein-Gordon equation in $1+1$ dimensions:
                $$left(frac{1}{c^2} partial_t^2-partial_x^2+mu Iright)u(t)=0$$
                This kind of notation is very useful, because the differential operator will take it's eigenvalue on it's eigenfunction, so you can easily get the dispersion relation, with just substitutions of the eigenvalues: $partial_t to (-i omega)$ and $partial_x to (ik)$ in the case of a plane wave.






                share|cite|improve this answer











                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  Robert did fully answer the question, but I'd like to add a different example: The differential equation of the harmonic oscillator is
                  $$frac{mathrm{d}^2 u(t)}{mathrm{d}t^2}+2beta frac{mathrm{d}u(t)}{mathrm{d}t}+omega^2 u(t)=0$$
                  And it can be rewritten as
                  $$(D^2+2beta D+omega^2 I)u(t)=0$$
                  Where $D$ is the differential operator: $Du(t)=frac{mathrm{d}u(t)}{mathrm{d}t}$.

                  Or the Klein-Gordon equation in $1+1$ dimensions:
                  $$left(frac{1}{c^2} partial_t^2-partial_x^2+mu Iright)u(t)=0$$
                  This kind of notation is very useful, because the differential operator will take it's eigenvalue on it's eigenfunction, so you can easily get the dispersion relation, with just substitutions of the eigenvalues: $partial_t to (-i omega)$ and $partial_x to (ik)$ in the case of a plane wave.






                  share|cite|improve this answer











                  $endgroup$



                  Robert did fully answer the question, but I'd like to add a different example: The differential equation of the harmonic oscillator is
                  $$frac{mathrm{d}^2 u(t)}{mathrm{d}t^2}+2beta frac{mathrm{d}u(t)}{mathrm{d}t}+omega^2 u(t)=0$$
                  And it can be rewritten as
                  $$(D^2+2beta D+omega^2 I)u(t)=0$$
                  Where $D$ is the differential operator: $Du(t)=frac{mathrm{d}u(t)}{mathrm{d}t}$.

                  Or the Klein-Gordon equation in $1+1$ dimensions:
                  $$left(frac{1}{c^2} partial_t^2-partial_x^2+mu Iright)u(t)=0$$
                  This kind of notation is very useful, because the differential operator will take it's eigenvalue on it's eigenfunction, so you can easily get the dispersion relation, with just substitutions of the eigenvalues: $partial_t to (-i omega)$ and $partial_x to (ik)$ in the case of a plane wave.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 10 '18 at 15:35

























                  answered Dec 10 '18 at 15:09









                  BotondBotond

                  5,7782732




                  5,7782732






























                      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%2f3033983%2fwhat-does-the-following-notation-mean-l-lambda-i%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

                      Probability when a professor distributes a quiz and homework assignment to a class of n students.

                      Image
                      0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
                      Read more

                      Aardman Animations

                      Image
                      Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
                      Read more

                      Are they similar matrix

                      Image
                      0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
                      Read more