Frobenius norm of Fourier matrix











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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







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  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    2 days ago










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    2 days ago












  • 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
    2 days ago










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    2 days ago















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






linear-algebra matrices matrix-calculus







share|cite|improve this question






















  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    2 days ago










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    2 days ago












  • 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
    2 days ago










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    2 days ago













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






linear-algebra matrices matrix-calculus







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




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 2 days ago









Studying Optimization

596




596












  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    2 days ago










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    2 days ago












  • 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
    2 days ago










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    2 days ago


















  • Do you know what’s the formula to compute the Frobenius norm?
    – lcv
    2 days ago










  • @lcv, yes I do. You can google it if you want to know
    – Studying Optimization
    2 days ago












  • 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
    2 days ago










  • @lcv, thanks, what I didnt see is that each entry squared has modulus one
    – Studying Optimization
    2 days ago
















Do you know what’s the formula to compute the Frobenius norm?
– lcv
2 days ago




Do you know what’s the formula to compute the Frobenius norm?
– lcv
2 days ago












@lcv, yes I do. You can google it if you want to know
– Studying Optimization
2 days ago






@lcv, yes I do. You can google it if you want to know
– Studying Optimization
2 days ago














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
2 days ago




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
2 days ago












@lcv, thanks, what I didnt see is that each entry squared has modulus one
– Studying Optimization
2 days ago




@lcv, thanks, what I didnt see is that each entry squared has modulus one
– Studying Optimization
2 days ago










1 Answer
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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





















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    up vote
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    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 2 days ago









        Omnomnomnom

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