Eigenvalues of an infinite matrix
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I have matrix elements
$$
H_{mn} =
begin{cases}
frac{1}{(m-n)^2} & m neq n \
0 & m=n
end{cases}$$
If the indices extend to infinity what are the eigenvalues of this matrix?
I'm sure this question is terribly formed, but I'm a physicist, so any suggestions on how to tackle this problem are appreciated.
matrices eigenvalues-eigenvectors infinity
asked Nov 15 at 15:26
Tom
214312
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up vote
0
down vote
favorite
I have matrix elements
$$
H_{mn} =
begin{cases}
frac{1}{(m-n)^2} & m neq n \
0 & m=n
end{cases}$$
If the indices extend to infinity what are the eigenvalues of this matrix?
I'm sure this question is terribly formed, but I'm a physicist, so any suggestions on how to tackle this problem are appreciated.
matrices eigenvalues-eigenvectors infinity
asked Nov 15 at 15:26
Tom
214312
The matrix is an infinite circulant (entries depend only on $m-n$). So it is similar to a diagonal matrix by Fourier transformation, i.e. look at $F^{-1}HF$ where $F$ is the discrete Fourier tfm matrix. Hence, if you write $c$ for the top row, find $Fc$. Then take an appropriate continuum limit ...
– Richard Martin
Nov 15 at 15:31
Are $m,n in mathbb{Z}$ (doubly infinite)?
– Hans Engler
Nov 15 at 15:33
@HansEngler yes
– Tom
Nov 15 at 15:37
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have matrix elements
$$
H_{mn} =
begin{cases}
frac{1}{(m-n)^2} & m neq n \
0 & m=n
end{cases}$$
If the indices extend to infinity what are the eigenvalues of this matrix?
I'm sure this question is terribly formed, but I'm a physicist, so any suggestions on how to tackle this problem are appreciated.
matrices eigenvalues-eigenvectors infinity
asked Nov 15 at 15:26
Tom
214312
I have matrix elements
$$
H_{mn} =
begin{cases}
frac{1}{(m-n)^2} & m neq n \
0 & m=n
end{cases}$$
If the indices extend to infinity what are the eigenvalues of this matrix?
I'm sure this question is terribly formed, but I'm a physicist, so any suggestions on how to tackle this problem are appreciated.
matrices eigenvalues-eigenvectors infinity
matrices eigenvalues-eigenvectors infinity
asked Nov 15 at 15:26
Tom
214312
asked Nov 15 at 15:26
Tom
214312
asked Nov 15 at 15:26
Tom
214312
asked Nov 15 at 15:26
Tom
214312
asked Nov 15 at 15:26
Tom
214312
214312
The matrix is an infinite circulant (entries depend only on $m-n$). So it is similar to a diagonal matrix by Fourier transformation, i.e. look at $F^{-1}HF$ where $F$ is the discrete Fourier tfm matrix. Hence, if you write $c$ for the top row, find $Fc$. Then take an appropriate continuum limit ...
– Richard Martin
Nov 15 at 15:31
Are $m,n in mathbb{Z}$ (doubly infinite)?
– Hans Engler
Nov 15 at 15:33
@HansEngler yes
– Tom
Nov 15 at 15:37
add a comment |
The matrix is an infinite circulant (entries depend only on $m-n$). So it is similar to a diagonal matrix by Fourier transformation, i.e. look at $F^{-1}HF$ where $F$ is the discrete Fourier tfm matrix. Hence, if you write $c$ for the top row, find $Fc$. Then take an appropriate continuum limit ...
– Richard Martin
Nov 15 at 15:31
Are $m,n in mathbb{Z}$ (doubly infinite)?
– Hans Engler
Nov 15 at 15:33
@HansEngler yes
– Tom
Nov 15 at 15:37
The matrix is an infinite circulant (entries depend only on $m-n$). So it is similar to a diagonal matrix by Fourier transformation, i.e. look at $F^{-1}HF$ where $F$ is the discrete Fourier tfm matrix. Hence, if you write $c$ for the top row, find $Fc$. Then take an appropriate continuum limit ...
– Richard Martin
Nov 15 at 15:31
The matrix is an infinite circulant (entries depend only on $m-n$). So it is similar to a diagonal matrix by Fourier transformation, i.e. look at $F^{-1}HF$ where $F$ is the discrete Fourier tfm matrix. Hence, if you write $c$ for the top row, find $Fc$. Then take an appropriate continuum limit ...
– Richard Martin
Nov 15 at 15:31
Are $m,n in mathbb{Z}$ (doubly infinite)?
– Hans Engler
Nov 15 at 15:33
Are $m,n in mathbb{Z}$ (doubly infinite)?
– Hans Engler
Nov 15 at 15:33
@HansEngler yes
– Tom
Nov 15 at 15:37
@HansEngler yes
– Tom
Nov 15 at 15:37
add a comment |
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The matrix is an infinite circulant (entries depend only on $m-n$). So it is similar to a diagonal matrix by Fourier transformation, i.e. look at $F^{-1}HF$ where $F$ is the discrete Fourier tfm matrix. Hence, if you write $c$ for the top row, find $Fc$. Then take an appropriate continuum limit ...
– Richard Martin
Nov 15 at 15:31
Are $m,n in mathbb{Z}$ (doubly infinite)?
– Hans Engler
Nov 15 at 15:33
@HansEngler yes
– Tom
Nov 15 at 15:37