Eigenvalues of random matrix
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I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows:
Let $A$ be random matrix s.t. $A=(a_{ij})$ and $a_{ij}sim N(0,sigma^{2})$. Let $lambda$ be eigenvalue of random matrix A and let $x$ be eigenvector of $A$. As
begin{equation}
Ax=lambda x
end{equation}
we can expand this as
begin{equation}
a_{i1}x_{1}+cdots+a_{in}x_{n}=lambda x_{i}
end{equation}
And IF WE ALLOW $x_{i}$ to be scalars not depending on random variables $a_{ij}$,
begin{align}
mathbb{E}(a_{i1}x_{1}+cdots+a_{in}x_{n})&=mathbb{E}(lambda x_{i})\
mathbb{E}(a_{i1})x_{1}+cdots+ mathbb{E}(a_{in})x_{1}&=mathbb{E}(lambda)x_{i}\
0&=mathbb{E}(lambda)x_{i}
end{align}
Thus, $mathbb{E}(lambda)=0$(assuming $x_{i}neq0$)
And by same method, we can show variance of $lambda$ to be $mathbb{V}(lambda)=nsigma^{2}$.
I think there is a serious problem in that I assumed $x_{i}$s to be scalar.
Is there any way to solve this problem?
Is there any depiction or theorem about eigenvalues of a random matrix defined as I defined?
linear-algebra probability matrices eigenvalues-eigenvectors random-matrices
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
$endgroup$
add a comment |
$begingroup$
I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows:
Let $A$ be random matrix s.t. $A=(a_{ij})$ and $a_{ij}sim N(0,sigma^{2})$. Let $lambda$ be eigenvalue of random matrix A and let $x$ be eigenvector of $A$. As
begin{equation}
Ax=lambda x
end{equation}
we can expand this as
begin{equation}
a_{i1}x_{1}+cdots+a_{in}x_{n}=lambda x_{i}
end{equation}
And IF WE ALLOW $x_{i}$ to be scalars not depending on random variables $a_{ij}$,
begin{align}
mathbb{E}(a_{i1}x_{1}+cdots+a_{in}x_{n})&=mathbb{E}(lambda x_{i})\
mathbb{E}(a_{i1})x_{1}+cdots+ mathbb{E}(a_{in})x_{1}&=mathbb{E}(lambda)x_{i}\
0&=mathbb{E}(lambda)x_{i}
end{align}
Thus, $mathbb{E}(lambda)=0$(assuming $x_{i}neq0$)
And by same method, we can show variance of $lambda$ to be $mathbb{V}(lambda)=nsigma^{2}$.
I think there is a serious problem in that I assumed $x_{i}$s to be scalar.
Is there any way to solve this problem?
Is there any depiction or theorem about eigenvalues of a random matrix defined as I defined?
linear-algebra probability matrices eigenvalues-eigenvectors random-matrices
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
$endgroup$
2
$begingroup$
Your problem is not well-posed. $A$ has $n$ random eigenvalues. You need to specify a method for picking one of them before talking about its expected value.
$endgroup$
– user1551
Dec 8 '18 at 9:47
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Wigner's Semicircle Law might be of interest to you.
$endgroup$
– JimmyK4542
Dec 8 '18 at 9:50
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@user1551 What can be said about maximum eigenvalue of $A$??
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:01
$begingroup$
@JimmyK4542 Thx very much!
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:04
$begingroup$
$mathbb C$ is not an ordered field. There is no such thing as a "maximum" eigenvalue. If you are talking about the spectral radius, I don't know the answer, but I'd bet that this has been studied in the literature.
$endgroup$
– user1551
Dec 8 '18 at 10:10
add a comment |
$begingroup$
I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows:
Let $A$ be random matrix s.t. $A=(a_{ij})$ and $a_{ij}sim N(0,sigma^{2})$. Let $lambda$ be eigenvalue of random matrix A and let $x$ be eigenvector of $A$. As
begin{equation}
Ax=lambda x
end{equation}
we can expand this as
begin{equation}
a_{i1}x_{1}+cdots+a_{in}x_{n}=lambda x_{i}
end{equation}
And IF WE ALLOW $x_{i}$ to be scalars not depending on random variables $a_{ij}$,
begin{align}
mathbb{E}(a_{i1}x_{1}+cdots+a_{in}x_{n})&=mathbb{E}(lambda x_{i})\
mathbb{E}(a_{i1})x_{1}+cdots+ mathbb{E}(a_{in})x_{1}&=mathbb{E}(lambda)x_{i}\
0&=mathbb{E}(lambda)x_{i}
end{align}
Thus, $mathbb{E}(lambda)=0$(assuming $x_{i}neq0$)
And by same method, we can show variance of $lambda$ to be $mathbb{V}(lambda)=nsigma^{2}$.
I think there is a serious problem in that I assumed $x_{i}$s to be scalar.
Is there any way to solve this problem?
Is there any depiction or theorem about eigenvalues of a random matrix defined as I defined?
linear-algebra probability matrices eigenvalues-eigenvectors random-matrices
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
$endgroup$
I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows:
Let $A$ be random matrix s.t. $A=(a_{ij})$ and $a_{ij}sim N(0,sigma^{2})$. Let $lambda$ be eigenvalue of random matrix A and let $x$ be eigenvector of $A$. As
begin{equation}
Ax=lambda x
end{equation}
we can expand this as
begin{equation}
a_{i1}x_{1}+cdots+a_{in}x_{n}=lambda x_{i}
end{equation}
And IF WE ALLOW $x_{i}$ to be scalars not depending on random variables $a_{ij}$,
begin{align}
mathbb{E}(a_{i1}x_{1}+cdots+a_{in}x_{n})&=mathbb{E}(lambda x_{i})\
mathbb{E}(a_{i1})x_{1}+cdots+ mathbb{E}(a_{in})x_{1}&=mathbb{E}(lambda)x_{i}\
0&=mathbb{E}(lambda)x_{i}
end{align}
Thus, $mathbb{E}(lambda)=0$(assuming $x_{i}neq0$)
And by same method, we can show variance of $lambda$ to be $mathbb{V}(lambda)=nsigma^{2}$.
I think there is a serious problem in that I assumed $x_{i}$s to be scalar.
Is there any way to solve this problem?
Is there any depiction or theorem about eigenvalues of a random matrix defined as I defined?
linear-algebra probability matrices eigenvalues-eigenvectors random-matrices
linear-algebra probability matrices eigenvalues-eigenvectors random-matrices
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
edited Dec 8 '18 at 10:06
Ryoungwoo Jang
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
asked Dec 8 '18 at 9:38
Ryoungwoo JangRyoungwoo Jang
35327
35327
2
$begingroup$
Your problem is not well-posed. $A$ has $n$ random eigenvalues. You need to specify a method for picking one of them before talking about its expected value.
$endgroup$
– user1551
Dec 8 '18 at 9:47
$begingroup$
Wigner's Semicircle Law might be of interest to you.
$endgroup$
– JimmyK4542
Dec 8 '18 at 9:50
$begingroup$
@user1551 What can be said about maximum eigenvalue of $A$??
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:01
$begingroup$
@JimmyK4542 Thx very much!
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:04
$begingroup$
$mathbb C$ is not an ordered field. There is no such thing as a "maximum" eigenvalue. If you are talking about the spectral radius, I don't know the answer, but I'd bet that this has been studied in the literature.
$endgroup$
– user1551
Dec 8 '18 at 10:10
add a comment |
2
$begingroup$
Your problem is not well-posed. $A$ has $n$ random eigenvalues. You need to specify a method for picking one of them before talking about its expected value.
$endgroup$
– user1551
Dec 8 '18 at 9:47
$begingroup$
Wigner's Semicircle Law might be of interest to you.
$endgroup$
– JimmyK4542
Dec 8 '18 at 9:50
$begingroup$
@user1551 What can be said about maximum eigenvalue of $A$??
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:01
$begingroup$
@JimmyK4542 Thx very much!
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:04
$begingroup$
$mathbb C$ is not an ordered field. There is no such thing as a "maximum" eigenvalue. If you are talking about the spectral radius, I don't know the answer, but I'd bet that this has been studied in the literature.
$endgroup$
– user1551
Dec 8 '18 at 10:10
2
2
$begingroup$
Your problem is not well-posed. $A$ has $n$ random eigenvalues. You need to specify a method for picking one of them before talking about its expected value.
$endgroup$
– user1551
Dec 8 '18 at 9:47
$begingroup$
Your problem is not well-posed. $A$ has $n$ random eigenvalues. You need to specify a method for picking one of them before talking about its expected value.
$endgroup$
– user1551
Dec 8 '18 at 9:47
$begingroup$
Wigner's Semicircle Law might be of interest to you.
$endgroup$
– JimmyK4542
Dec 8 '18 at 9:50
$begingroup$
Wigner's Semicircle Law might be of interest to you.
$endgroup$
– JimmyK4542
Dec 8 '18 at 9:50
$begingroup$
@user1551 What can be said about maximum eigenvalue of $A$??
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:01
$begingroup$
@user1551 What can be said about maximum eigenvalue of $A$??
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:01
$begingroup$
@JimmyK4542 Thx very much!
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:04
$begingroup$
@JimmyK4542 Thx very much!
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:04
$begingroup$
$mathbb C$ is not an ordered field. There is no such thing as a "maximum" eigenvalue. If you are talking about the spectral radius, I don't know the answer, but I'd bet that this has been studied in the literature.
$endgroup$
– user1551
Dec 8 '18 at 10:10
$begingroup$
$mathbb C$ is not an ordered field. There is no such thing as a "maximum" eigenvalue. If you are talking about the spectral radius, I don't know the answer, but I'd bet that this has been studied in the literature.
$endgroup$
– user1551
Dec 8 '18 at 10:10
add a comment |
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$begingroup$
Your problem is not well-posed. $A$ has $n$ random eigenvalues. You need to specify a method for picking one of them before talking about its expected value.
$endgroup$
– user1551
Dec 8 '18 at 9:47
$begingroup$
Wigner's Semicircle Law might be of interest to you.
$endgroup$
– JimmyK4542
Dec 8 '18 at 9:50
$begingroup$
@user1551 What can be said about maximum eigenvalue of $A$??
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:01
$begingroup$
@JimmyK4542 Thx very much!
$endgroup$
– Ryoungwoo Jang
Dec 8 '18 at 10:04
$begingroup$
$mathbb C$ is not an ordered field. There is no such thing as a "maximum" eigenvalue. If you are talking about the spectral radius, I don't know the answer, but I'd bet that this has been studied in the literature.
$endgroup$
– user1551
Dec 8 '18 at 10:10