Frobenius norm of Fourier matrix











up vote
0
down vote

favorite
1












Fourier matrix is given as enter image description here



where $omega = e^{-2pi i/N}$



Is there any clever way to calculate Frobenius norm of Fourier matrix? I tried solving it with brute force and got some ugly calculations










share|cite|improve this question






















  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    Nov 19 at 18:55










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    Nov 19 at 18:57












  • Thank you 😊. So you only need to compute the sum of the absolute values squared of all the entries. Note that each entry has modulus one. Can you take it from here?
    – lcv
    Nov 19 at 19:00










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    Nov 19 at 19:05















up vote
0
down vote

favorite
1












Fourier matrix is given as enter image description here



where $omega = e^{-2pi i/N}$



Is there any clever way to calculate Frobenius norm of Fourier matrix? I tried solving it with brute force and got some ugly calculations










share|cite|improve this question






















  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    Nov 19 at 18:55










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    Nov 19 at 18:57












  • Thank you 😊. So you only need to compute the sum of the absolute values squared of all the entries. Note that each entry has modulus one. Can you take it from here?
    – lcv
    Nov 19 at 19:00










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    Nov 19 at 19:05













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Fourier matrix is given as enter image description here



where $omega = e^{-2pi i/N}$



Is there any clever way to calculate Frobenius norm of Fourier matrix? I tried solving it with brute force and got some ugly calculations










share|cite|improve this question













Fourier matrix is given as enter image description here



where $omega = e^{-2pi i/N}$



Is there any clever way to calculate Frobenius norm of Fourier matrix? I tried solving it with brute force and got some ugly calculations







linear-algebra matrices matrix-calculus






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 19 at 18:43









Studying Optimization

626




626












  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    Nov 19 at 18:55










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    Nov 19 at 18:57












  • Thank you 😊. So you only need to compute the sum of the absolute values squared of all the entries. Note that each entry has modulus one. Can you take it from here?
    – lcv
    Nov 19 at 19:00










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    Nov 19 at 19:05


















  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    Nov 19 at 18:55










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    Nov 19 at 18:57












  • Thank you 😊. So you only need to compute the sum of the absolute values squared of all the entries. Note that each entry has modulus one. Can you take it from here?
    – lcv
    Nov 19 at 19:00










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    Nov 19 at 19:05
















Do you know what’s the formula to compute the Frobenius norm?
– lcv
Nov 19 at 18:55




Do you know what’s the formula to compute the Frobenius norm?
– lcv
Nov 19 at 18:55












@lcv, yes I do. You can google it if you want to know
– Studying Optimization
Nov 19 at 18:57






@lcv, yes I do. You can google it if you want to know
– Studying Optimization
Nov 19 at 18:57














Thank you 😊. So you only need to compute the sum of the absolute values squared of all the entries. Note that each entry has modulus one. Can you take it from here?
– lcv
Nov 19 at 19:00




Thank you 😊. So you only need to compute the sum of the absolute values squared of all the entries. Note that each entry has modulus one. Can you take it from here?
– lcv
Nov 19 at 19:00












@lcv, thanks, what I didnt see is that each entry squared has modulus one
– Studying Optimization
Nov 19 at 19:05




@lcv, thanks, what I didnt see is that each entry squared has modulus one
– Studying Optimization
Nov 19 at 19:05










1 Answer
1






active

oldest

votes

















up vote
0
down vote



accepted










This is a straightforward computation using any reasonable definition of the Frobenius norm. For instance, we have
$$
|W| = sqrt{operatorname{tr}(W^*W)} = sqrt{operatorname{tr}(I)} = sqrt{N}
$$

where $W^*W = I$ since $W$ is a unitary matrix.






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
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005333%2ffrobenius-norm-of-fourier-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
    0
    down vote



    accepted










    This is a straightforward computation using any reasonable definition of the Frobenius norm. For instance, we have
    $$
    |W| = sqrt{operatorname{tr}(W^*W)} = sqrt{operatorname{tr}(I)} = sqrt{N}
    $$

    where $W^*W = I$ since $W$ is a unitary matrix.






    share|cite|improve this answer

























      up vote
      0
      down vote



      accepted










      This is a straightforward computation using any reasonable definition of the Frobenius norm. For instance, we have
      $$
      |W| = sqrt{operatorname{tr}(W^*W)} = sqrt{operatorname{tr}(I)} = sqrt{N}
      $$

      where $W^*W = I$ since $W$ is a unitary matrix.






      share|cite|improve this answer























        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        This is a straightforward computation using any reasonable definition of the Frobenius norm. For instance, we have
        $$
        |W| = sqrt{operatorname{tr}(W^*W)} = sqrt{operatorname{tr}(I)} = sqrt{N}
        $$

        where $W^*W = I$ since $W$ is a unitary matrix.






        share|cite|improve this answer












        This is a straightforward computation using any reasonable definition of the Frobenius norm. For instance, we have
        $$
        |W| = sqrt{operatorname{tr}(W^*W)} = sqrt{operatorname{tr}(I)} = sqrt{N}
        $$

        where $W^*W = I$ since $W$ is a unitary matrix.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 19 at 19:02









        Omnomnomnom

        125k788176




        125k788176






























            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%2f3005333%2ffrobenius-norm-of-fourier-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

            How to change which sound is reproduced for terminal bell?

            Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

            Can I use Tabulator js library in my java Spring + Thymeleaf project?