Connection between $operatorname{Var}(M^n v)$ and largest eigenvalue of $M$
$begingroup$
In a proof I am trying to understand, the following is stated:
$ M$ is a non-random matrix with eigenvalues $lambda_i$, $v$ is a random vector, $n$ is a scalar,
$operatorname{Var}(M^n v) ge max(|lambda_i|)^n
$
As I copied this from the blackboard, I might have made a mistake. Our script says
$ lim_{ntoinfty}operatorname{Var}(M^n v) = infty mbox{ if } max(|lambda_i|) > 1
$
(which would be implied by the first inequality).
If anyone could explain why this holds, or point me to a paper/book where this is explained I would very much appreciate it.
statistics eigenvalues-eigenvectors random-variables covariance variance
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
asked Dec 27 '18 at 12:10
CariieCariie
133
$endgroup$
add a comment |
$begingroup$
In a proof I am trying to understand, the following is stated:
$ M$ is a non-random matrix with eigenvalues $lambda_i$, $v$ is a random vector, $n$ is a scalar,
$operatorname{Var}(M^n v) ge max(|lambda_i|)^n
$
As I copied this from the blackboard, I might have made a mistake. Our script says
$ lim_{ntoinfty}operatorname{Var}(M^n v) = infty mbox{ if } max(|lambda_i|) > 1
$
(which would be implied by the first inequality).
If anyone could explain why this holds, or point me to a paper/book where this is explained I would very much appreciate it.
statistics eigenvalues-eigenvectors random-variables covariance variance
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
asked Dec 27 '18 at 12:10
CariieCariie
133
$endgroup$
add a comment |
$begingroup$
In a proof I am trying to understand, the following is stated:
$ M$ is a non-random matrix with eigenvalues $lambda_i$, $v$ is a random vector, $n$ is a scalar,
$operatorname{Var}(M^n v) ge max(|lambda_i|)^n
$
As I copied this from the blackboard, I might have made a mistake. Our script says
$ lim_{ntoinfty}operatorname{Var}(M^n v) = infty mbox{ if } max(|lambda_i|) > 1
$
(which would be implied by the first inequality).
If anyone could explain why this holds, or point me to a paper/book where this is explained I would very much appreciate it.
statistics eigenvalues-eigenvectors random-variables covariance variance
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
asked Dec 27 '18 at 12:10
CariieCariie
133
$endgroup$
In a proof I am trying to understand, the following is stated:
$ M$ is a non-random matrix with eigenvalues $lambda_i$, $v$ is a random vector, $n$ is a scalar,
$operatorname{Var}(M^n v) ge max(|lambda_i|)^n
$
As I copied this from the blackboard, I might have made a mistake. Our script says
$ lim_{ntoinfty}operatorname{Var}(M^n v) = infty mbox{ if } max(|lambda_i|) > 1
$
(which would be implied by the first inequality).
If anyone could explain why this holds, or point me to a paper/book where this is explained I would very much appreciate it.
statistics eigenvalues-eigenvectors random-variables covariance variance
statistics eigenvalues-eigenvectors random-variables covariance variance
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
asked Dec 27 '18 at 12:10
CariieCariie
133
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
asked Dec 27 '18 at 12:10
CariieCariie
133
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
edited Dec 31 '18 at 21:12
Davide Giraudo
128k17154268
128k17154268
asked Dec 27 '18 at 12:10
CariieCariie
133
asked Dec 27 '18 at 12:10
CariieCariie
133
asked Dec 27 '18 at 12:10
CariieCariie
133
133
add a comment |
add a comment |
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