Frobenius norm of Fourier matrix
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The Fourier matrix is given by
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
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
$endgroup$
add a comment |
$begingroup$
The Fourier matrix is given by
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
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
$endgroup$
$begingroup$
Do you know what’s the formula to compute the Frobenius norm?
$endgroup$
– lcv
Nov 19 '18 at 18:55
$begingroup$
@lcv, yes I do. You can google it if you want to know
$endgroup$
– Studying Optimization
Nov 19 '18 at 18:57
$begingroup$
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?
$endgroup$
– lcv
Nov 19 '18 at 19:00
$begingroup$
@lcv, thanks, what I didnt see is that each entry squared has modulus one
$endgroup$
– Studying Optimization
Nov 19 '18 at 19:05
add a comment |
$begingroup$
The Fourier matrix is given by
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
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
$endgroup$
The Fourier matrix is given by
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
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
edited Jan 22 at 7:56
Rodrigo de Azevedo
13.1k41959
13.1k41959
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
asked Nov 19 '18 at 18:43
Studying OptimizationStudying Optimization
867
867
$begingroup$
Do you know what’s the formula to compute the Frobenius norm?
$endgroup$
– lcv
Nov 19 '18 at 18:55
$begingroup$
@lcv, yes I do. You can google it if you want to know
$endgroup$
– Studying Optimization
Nov 19 '18 at 18:57
$begingroup$
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?
$endgroup$
– lcv
Nov 19 '18 at 19:00
$begingroup$
@lcv, thanks, what I didnt see is that each entry squared has modulus one
$endgroup$
– Studying Optimization
Nov 19 '18 at 19:05
add a comment |
$begingroup$
Do you know what’s the formula to compute the Frobenius norm?
$endgroup$
– lcv
Nov 19 '18 at 18:55
$begingroup$
@lcv, yes I do. You can google it if you want to know
$endgroup$
– Studying Optimization
Nov 19 '18 at 18:57
$begingroup$
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?
$endgroup$
– lcv
Nov 19 '18 at 19:00
$begingroup$
@lcv, thanks, what I didnt see is that each entry squared has modulus one
$endgroup$
– Studying Optimization
Nov 19 '18 at 19:05
$begingroup$
Do you know what’s the formula to compute the Frobenius norm?
$endgroup$
– lcv
Nov 19 '18 at 18:55
$begingroup$
Do you know what’s the formula to compute the Frobenius norm?
$endgroup$
– lcv
Nov 19 '18 at 18:55
$begingroup$
@lcv, yes I do. You can google it if you want to know
$endgroup$
– Studying Optimization
Nov 19 '18 at 18:57
$begingroup$
@lcv, yes I do. You can google it if you want to know
$endgroup$
– Studying Optimization
Nov 19 '18 at 18:57
$begingroup$
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?
$endgroup$
– lcv
Nov 19 '18 at 19:00
$begingroup$
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?
$endgroup$
– lcv
Nov 19 '18 at 19:00
$begingroup$
@lcv, thanks, what I didnt see is that each entry squared has modulus one
$endgroup$
– Studying Optimization
Nov 19 '18 at 19:05
$begingroup$
@lcv, thanks, what I didnt see is that each entry squared has modulus one
$endgroup$
– Studying Optimization
Nov 19 '18 at 19:05
add a comment |
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.
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
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.
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
$endgroup$
add a comment |
$begingroup$
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.
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
$endgroup$
add a comment |
$begingroup$
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.
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
$endgroup$
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.
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
answered Nov 19 '18 at 19:02
OmnomnomnomOmnomnomnom
128k791184
128k791184
add a comment |
add a comment |
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$begingroup$
Do you know what’s the formula to compute the Frobenius norm?
$endgroup$
– lcv
Nov 19 '18 at 18:55
$begingroup$
@lcv, yes I do. You can google it if you want to know
$endgroup$
– Studying Optimization
Nov 19 '18 at 18:57
$begingroup$
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?
$endgroup$
– lcv
Nov 19 '18 at 19:00
$begingroup$
@lcv, thanks, what I didnt see is that each entry squared has modulus one
$endgroup$
– Studying Optimization
Nov 19 '18 at 19:05