Symplectic forms are isomorphic

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
3
down vote

favorite












Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem:




All symplectic forms on $V$ are ismorphic.




I have two questions about this:



1) Can somebody explain what it means precisely that two symplectic forms are isomorphic?



2) Could somebode give references on this statement with or without proof?



Thank you very much.




linear-algebra abstract-algebra reference-request symplectic-linear-algebra




share|cite|improve this question
























    up vote
    3
    down vote

    favorite












    Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem:




    All symplectic forms on $V$ are ismorphic.




    I have two questions about this:



    1) Can somebody explain what it means precisely that two symplectic forms are isomorphic?



    2) Could somebode give references on this statement with or without proof?



    Thank you very much.




    linear-algebra abstract-algebra reference-request symplectic-linear-algebra




    share|cite|improve this question






















      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem:




      All symplectic forms on $V$ are ismorphic.




      I have two questions about this:



      1) Can somebody explain what it means precisely that two symplectic forms are isomorphic?



      2) Could somebode give references on this statement with or without proof?



      Thank you very much.




      linear-algebra abstract-algebra reference-request symplectic-linear-algebra




      share|cite|improve this question












      Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem:




      All symplectic forms on $V$ are ismorphic.




      I have two questions about this:



      1) Can somebody explain what it means precisely that two symplectic forms are isomorphic?



      2) Could somebode give references on this statement with or without proof?



      Thank you very much.





      linear-algebra abstract-algebra reference-request symplectic-linear-algebra





      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 27 at 15:42







      user586527



























          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $omega$ has the form:



          $$beginpmatrix
          0 & I_n \
          -I_n & 0
          endpmatrix$$



          This proves that two symplectic forms are isomorphic. A reference is Wikipedia.






          share|cite|improve this answer




















          • Merci! And what does it mean if two such forms are isomorphic?
            – user586527
            Aug 27 at 16:20










          • It means that for any $omega_1,omega_2$ two symplectic forms, you can find an invertible linear transformation $varphi$ such that for any $x,y in V$ $omega_1(x,y) = omega_2(varphi(x),varphi(y))$.
            – mathcounterexamples.net
            Aug 27 at 16:24










          • So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well?
            – user586527
            Aug 27 at 16:46










          • Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general.
            – mathcounterexamples.net
            Aug 27 at 16:49










          • What does ate mean?
            – user586527
            Aug 27 at 16:50










          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: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          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%2f2896320%2fsymplectic-forms-are-isomorphic%23new-answer', 'question_page');

          );

          Post as a guest
















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote



          accepted










          It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $omega$ has the form:



          $$beginpmatrix
          0 & I_n \
          -I_n & 0
          endpmatrix$$



          This proves that two symplectic forms are isomorphic. A reference is Wikipedia.






          share|cite|improve this answer




















          • Merci! And what does it mean if two such forms are isomorphic?
            – user586527
            Aug 27 at 16:20










          • It means that for any $omega_1,omega_2$ two symplectic forms, you can find an invertible linear transformation $varphi$ such that for any $x,y in V$ $omega_1(x,y) = omega_2(varphi(x),varphi(y))$.
            – mathcounterexamples.net
            Aug 27 at 16:24










          • So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well?
            – user586527
            Aug 27 at 16:46










          • Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general.
            – mathcounterexamples.net
            Aug 27 at 16:49










          • What does ate mean?
            – user586527
            Aug 27 at 16:50














          up vote
          1
          down vote



          accepted










          It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $omega$ has the form:



          $$beginpmatrix
          0 & I_n \
          -I_n & 0
          endpmatrix$$



          This proves that two symplectic forms are isomorphic. A reference is Wikipedia.






          share|cite|improve this answer




















          • Merci! And what does it mean if two such forms are isomorphic?
            – user586527
            Aug 27 at 16:20










          • It means that for any $omega_1,omega_2$ two symplectic forms, you can find an invertible linear transformation $varphi$ such that for any $x,y in V$ $omega_1(x,y) = omega_2(varphi(x),varphi(y))$.
            – mathcounterexamples.net
            Aug 27 at 16:24










          • So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well?
            – user586527
            Aug 27 at 16:46










          • Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general.
            – mathcounterexamples.net
            Aug 27 at 16:49










          • What does ate mean?
            – user586527
            Aug 27 at 16:50












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $omega$ has the form:



          $$beginpmatrix
          0 & I_n \
          -I_n & 0
          endpmatrix$$



          This proves that two symplectic forms are isomorphic. A reference is Wikipedia.






          share|cite|improve this answer












          It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $omega$ has the form:



          $$beginpmatrix
          0 & I_n \
          -I_n & 0
          endpmatrix$$



          This proves that two symplectic forms are isomorphic. A reference is Wikipedia.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 27 at 16:19









          mathcounterexamples.net

          25.5k21754




          25.5k21754











          • Merci! And what does it mean if two such forms are isomorphic?
            – user586527
            Aug 27 at 16:20










          • It means that for any $omega_1,omega_2$ two symplectic forms, you can find an invertible linear transformation $varphi$ such that for any $x,y in V$ $omega_1(x,y) = omega_2(varphi(x),varphi(y))$.
            – mathcounterexamples.net
            Aug 27 at 16:24










          • So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well?
            – user586527
            Aug 27 at 16:46










          • Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general.
            – mathcounterexamples.net
            Aug 27 at 16:49










          • What does ate mean?
            – user586527
            Aug 27 at 16:50
















          • Merci! And what does it mean if two such forms are isomorphic?
            – user586527
            Aug 27 at 16:20










          • It means that for any $omega_1,omega_2$ two symplectic forms, you can find an invertible linear transformation $varphi$ such that for any $x,y in V$ $omega_1(x,y) = omega_2(varphi(x),varphi(y))$.
            – mathcounterexamples.net
            Aug 27 at 16:24










          • So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well?
            – user586527
            Aug 27 at 16:46










          • Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general.
            – mathcounterexamples.net
            Aug 27 at 16:49










          • What does ate mean?
            – user586527
            Aug 27 at 16:50















          Merci! And what does it mean if two such forms are isomorphic?
          – user586527
          Aug 27 at 16:20




          Merci! And what does it mean if two such forms are isomorphic?
          – user586527
          Aug 27 at 16:20












          It means that for any $omega_1,omega_2$ two symplectic forms, you can find an invertible linear transformation $varphi$ such that for any $x,y in V$ $omega_1(x,y) = omega_2(varphi(x),varphi(y))$.
          – mathcounterexamples.net
          Aug 27 at 16:24




          It means that for any $omega_1,omega_2$ two symplectic forms, you can find an invertible linear transformation $varphi$ such that for any $x,y in V$ $omega_1(x,y) = omega_2(varphi(x),varphi(y))$.
          – mathcounterexamples.net
          Aug 27 at 16:24












          So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well?
          – user586527
          Aug 27 at 16:46




          So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well?
          – user586527
          Aug 27 at 16:46












          Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general.
          – mathcounterexamples.net
          Aug 27 at 16:49




          Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general.
          – mathcounterexamples.net
          Aug 27 at 16:49












          What does ate mean?
          – user586527
          Aug 27 at 16:50




          What does ate mean?
          – user586527
          Aug 27 at 16:50

















           

          draft saved


          draft discarded















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2896320%2fsymplectic-forms-are-isomorphic%23new-answer', 'question_page');

          );

          Post as a guest
































































          -

          這個網誌中的熱門文章

          How to combine Bézier curves to a surface?

          圖片
          Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
          閱讀完整內容

          Mutual Information Always Non-negative

          圖片
          Clash Royale CLAN TAG #URR8PPP up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 '12 at 15:38 Cameron Buie 83.6k 7 71 154 asked Jun 17 '12 at 15:30 Omri 359 3 13 3 Convexity of the function $tmapsto tlog t$. – Did Jun 17 '12 at 15:33 In addition, the convexity properties require the coefficients in the linear combination sum 1. Then, as p(x,y) is a probability distribution, it fullfits such condition. – Francisco Javier Delgado Ceped Aug 10 at 18:44 add a comment  |  up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 ...
          閱讀完整內容

          Why am i infinitely getting the same tweet with the Twitter Search API?

          圖片
          up vote 0 down vote favorite My code workes perfectly for other queries, but for one query I am infinitely getting the same tweet. I can't understand what the problem is. API call: query='Allergic asthma OR Nonallergic asthma OR Occupational asthma OR EIB OR Exercise-induced bronchoconstriction OR Nocturnal asthma OR Cough-variant asthma' new_tweets = api.search(q=searchQuery, count=100, since_id=since_id, lang='en',tweet_mode='extended') if not new_tweets: print("No more tweets found") break for tweet in new_tweets: if True: print(jsonpickle.encode(tweet._json, unpicklable=False)) my_file = open('searchTweets.txt','a') my_file.write(jsonpickle.encode(tweet._json, unpicklable=False)+'n') my_file.close() else: continue Output in file: https://pastebin.com/JmRCsTeE These are the JSONs of the same Tweet. Other queries that work: Breast cancer OR Prostate cancer OR Colon cancer OR Lung cancer Type 2 dia...
          閱讀完整內容