The defining matrix of a symplectic matrix











up vote
0
down vote

favorite












Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me.
A $2ntimes 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition:
$$M^TOmega M=Omega$$
where $Omega$ is a fixed $2ntimes 2n$ real, invertible and skew-symmetric matrix.



My question is: since $Omega$ can be arbitrary, so if $Omega,Delta$ are both satisfy the condition, then the following statement must be true:
$$M^TOmega M=Omega Rightarrow M^TDelta M=Delta.$$



But I don't know how to prove this. Can anyone help me? Thanks.










share|cite|improve this question







New contributor




Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
























    up vote
    0
    down vote

    favorite












    Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me.
    A $2ntimes 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition:
    $$M^TOmega M=Omega$$
    where $Omega$ is a fixed $2ntimes 2n$ real, invertible and skew-symmetric matrix.



    My question is: since $Omega$ can be arbitrary, so if $Omega,Delta$ are both satisfy the condition, then the following statement must be true:
    $$M^TOmega M=Omega Rightarrow M^TDelta M=Delta.$$



    But I don't know how to prove this. Can anyone help me? Thanks.










    share|cite|improve this question







    New contributor




    Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me.
      A $2ntimes 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition:
      $$M^TOmega M=Omega$$
      where $Omega$ is a fixed $2ntimes 2n$ real, invertible and skew-symmetric matrix.



      My question is: since $Omega$ can be arbitrary, so if $Omega,Delta$ are both satisfy the condition, then the following statement must be true:
      $$M^TOmega M=Omega Rightarrow M^TDelta M=Delta.$$



      But I don't know how to prove this. Can anyone help me? Thanks.










      share|cite|improve this question







      New contributor




      Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me.
      A $2ntimes 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition:
      $$M^TOmega M=Omega$$
      where $Omega$ is a fixed $2ntimes 2n$ real, invertible and skew-symmetric matrix.



      My question is: since $Omega$ can be arbitrary, so if $Omega,Delta$ are both satisfy the condition, then the following statement must be true:
      $$M^TOmega M=Omega Rightarrow M^TDelta M=Delta.$$



      But I don't know how to prove this. Can anyone help me? Thanks.







      linear-algebra symplectic-geometry symplectic-linear-algebra






      share|cite|improve this question







      New contributor




      Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question






      New contributor




      Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked Nov 15 at 9:30









      Arc Walker

      11




      11




      New contributor




      Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      Arc Walker is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          2
          down vote













          There are more than one symplectic structure oon a vector space, but they are isomorphic not equal, there exists a linear invertible map such that $fcircDelta =Omegacirc f$ where $Omega$ and $Delta$ are the linear map associated to the corresponding matrices.






          share|cite|improve this answer





















            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',
            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
            });


            }
            });






            Arc Walker is a new contributor. Be nice, and check out our Code of Conduct.










             

            draft saved


            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999461%2fthe-defining-matrix-of-a-symplectic-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








            up vote
            2
            down vote













            There are more than one symplectic structure oon a vector space, but they are isomorphic not equal, there exists a linear invertible map such that $fcircDelta =Omegacirc f$ where $Omega$ and $Delta$ are the linear map associated to the corresponding matrices.






            share|cite|improve this answer

























              up vote
              2
              down vote













              There are more than one symplectic structure oon a vector space, but they are isomorphic not equal, there exists a linear invertible map such that $fcircDelta =Omegacirc f$ where $Omega$ and $Delta$ are the linear map associated to the corresponding matrices.






              share|cite|improve this answer























                up vote
                2
                down vote










                up vote
                2
                down vote









                There are more than one symplectic structure oon a vector space, but they are isomorphic not equal, there exists a linear invertible map such that $fcircDelta =Omegacirc f$ where $Omega$ and $Delta$ are the linear map associated to the corresponding matrices.






                share|cite|improve this answer












                There are more than one symplectic structure oon a vector space, but they are isomorphic not equal, there exists a linear invertible map such that $fcircDelta =Omegacirc f$ where $Omega$ and $Delta$ are the linear map associated to the corresponding matrices.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 15 at 9:33









                Tsemo Aristide

                54.3k11344




                54.3k11344






















                    Arc Walker is a new contributor. Be nice, and check out our Code of Conduct.










                     

                    draft saved


                    draft discarded


















                    Arc Walker is a new contributor. Be nice, and check out our Code of Conduct.













                    Arc Walker is a new contributor. Be nice, and check out our Code of Conduct.












                    Arc Walker is a new contributor. Be nice, and check out our Code of Conduct.















                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999461%2fthe-defining-matrix-of-a-symplectic-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

                    AnyDesk - Fatal Program Failure

                    How to calibrate 16:9 built-in touch-screen to a 4:3 resolution?

                    QoS: MAC-Priority for clients behind a repeater