Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm...
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We need a help to find a reasonable condition such that the spectral radius for a special matrix $mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}$ is smaller than 1. Here $otimes$ is tensor product. These matrices $bf J$, $hat{bf{G}}$, $hat{bf{H}}$ are defined below. I am thinking about if there is a matrix norm $||$ such that $|mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}| < 1$, because spectral radius is smaller than any matrix norm.
$bf {J}$ is a general real-valued square matrix. We are NOT allowed to assume $bf J$ is some special type of matrix such as "symmetric matrix", "positive-definite matrix" or "non-negative matrix". $bf I$ is an identity matrix of the same size as $bf J$.
Let $hat{bf H}$ be a $phat{N}times p hat{N}$ sparse matrix consisting of $ptimes p$ blocks, where each block is of size $hat{N}timeshat{N}$. The values in $hat{bf H}$ is illustrated below (sorry I missed the coefficient before):
Let $hat{mathbf{G}}=begin{pmatrix}
{bf G} & & & & \
& ddots & & & \
& & {bf G} & & \
& & & ddots & \
& & & & {bf G} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal block matrix repeating $p$ times of $bf G$, which is an $hat{N} times hat{N}$ matrix defined as the following:
where $h$ is some positive coefficient. The eigenvalues of ${bf G}$ has analytical form $frac{h}{{{rm{2}}(cos (frac{{kpi }}{{widehat N + 1}}) - 1)}}$ where $k = 1,...,hat{N}$.
Let $hat{mathbf{W}}=begin{pmatrix}
{bf W} & & & & \
& ddots & & & \
& & {bf W} & & \
& & & ddots & \
& & & & {bf W} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal weight matrix repeating $p$ times of $bf {W}$, which is a diagonal weight matrix where all diagonal elements are non-negative and sum to $1$. The weights are free to choose, therefore we can simple choices like every element in $bf W$ is $frac{1}{hat{N}}$
linear-algebra eigenvalues-eigenvectors norm spectral-radius
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
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add a comment |
$begingroup$
We need a help to find a reasonable condition such that the spectral radius for a special matrix $mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}$ is smaller than 1. Here $otimes$ is tensor product. These matrices $bf J$, $hat{bf{G}}$, $hat{bf{H}}$ are defined below. I am thinking about if there is a matrix norm $||$ such that $|mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}| < 1$, because spectral radius is smaller than any matrix norm.
$bf {J}$ is a general real-valued square matrix. We are NOT allowed to assume $bf J$ is some special type of matrix such as "symmetric matrix", "positive-definite matrix" or "non-negative matrix". $bf I$ is an identity matrix of the same size as $bf J$.
Let $hat{bf H}$ be a $phat{N}times p hat{N}$ sparse matrix consisting of $ptimes p$ blocks, where each block is of size $hat{N}timeshat{N}$. The values in $hat{bf H}$ is illustrated below (sorry I missed the coefficient before):
Let $hat{mathbf{G}}=begin{pmatrix}
{bf G} & & & & \
& ddots & & & \
& & {bf G} & & \
& & & ddots & \
& & & & {bf G} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal block matrix repeating $p$ times of $bf G$, which is an $hat{N} times hat{N}$ matrix defined as the following:
where $h$ is some positive coefficient. The eigenvalues of ${bf G}$ has analytical form $frac{h}{{{rm{2}}(cos (frac{{kpi }}{{widehat N + 1}}) - 1)}}$ where $k = 1,...,hat{N}$.
Let $hat{mathbf{W}}=begin{pmatrix}
{bf W} & & & & \
& ddots & & & \
& & {bf W} & & \
& & & ddots & \
& & & & {bf W} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal weight matrix repeating $p$ times of $bf {W}$, which is a diagonal weight matrix where all diagonal elements are non-negative and sum to $1$. The weights are free to choose, therefore we can simple choices like every element in $bf W$ is $frac{1}{hat{N}}$
linear-algebra eigenvalues-eigenvectors norm spectral-radius
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
$endgroup$
add a comment |
$begingroup$
We need a help to find a reasonable condition such that the spectral radius for a special matrix $mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}$ is smaller than 1. Here $otimes$ is tensor product. These matrices $bf J$, $hat{bf{G}}$, $hat{bf{H}}$ are defined below. I am thinking about if there is a matrix norm $||$ such that $|mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}| < 1$, because spectral radius is smaller than any matrix norm.
$bf {J}$ is a general real-valued square matrix. We are NOT allowed to assume $bf J$ is some special type of matrix such as "symmetric matrix", "positive-definite matrix" or "non-negative matrix". $bf I$ is an identity matrix of the same size as $bf J$.
Let $hat{bf H}$ be a $phat{N}times p hat{N}$ sparse matrix consisting of $ptimes p$ blocks, where each block is of size $hat{N}timeshat{N}$. The values in $hat{bf H}$ is illustrated below (sorry I missed the coefficient before):
Let $hat{mathbf{G}}=begin{pmatrix}
{bf G} & & & & \
& ddots & & & \
& & {bf G} & & \
& & & ddots & \
& & & & {bf G} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal block matrix repeating $p$ times of $bf G$, which is an $hat{N} times hat{N}$ matrix defined as the following:
where $h$ is some positive coefficient. The eigenvalues of ${bf G}$ has analytical form $frac{h}{{{rm{2}}(cos (frac{{kpi }}{{widehat N + 1}}) - 1)}}$ where $k = 1,...,hat{N}$.
Let $hat{mathbf{W}}=begin{pmatrix}
{bf W} & & & & \
& ddots & & & \
& & {bf W} & & \
& & & ddots & \
& & & & {bf W} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal weight matrix repeating $p$ times of $bf {W}$, which is a diagonal weight matrix where all diagonal elements are non-negative and sum to $1$. The weights are free to choose, therefore we can simple choices like every element in $bf W$ is $frac{1}{hat{N}}$
linear-algebra eigenvalues-eigenvectors norm spectral-radius
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
$endgroup$
We need a help to find a reasonable condition such that the spectral radius for a special matrix $mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}$ is smaller than 1. Here $otimes$ is tensor product. These matrices $bf J$, $hat{bf{G}}$, $hat{bf{H}}$ are defined below. I am thinking about if there is a matrix norm $||$ such that $|mathbf{J} otimeshat{mathbf{G}}hat{mathbf{W}} + mathbf{I}otimesmathbf{hat{H}}| < 1$, because spectral radius is smaller than any matrix norm.
$bf {J}$ is a general real-valued square matrix. We are NOT allowed to assume $bf J$ is some special type of matrix such as "symmetric matrix", "positive-definite matrix" or "non-negative matrix". $bf I$ is an identity matrix of the same size as $bf J$.
Let $hat{bf H}$ be a $phat{N}times p hat{N}$ sparse matrix consisting of $ptimes p$ blocks, where each block is of size $hat{N}timeshat{N}$. The values in $hat{bf H}$ is illustrated below (sorry I missed the coefficient before):
Let $hat{mathbf{G}}=begin{pmatrix}
{bf G} & & & & \
& ddots & & & \
& & {bf G} & & \
& & & ddots & \
& & & & {bf G} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal block matrix repeating $p$ times of $bf G$, which is an $hat{N} times hat{N}$ matrix defined as the following:
where $h$ is some positive coefficient. The eigenvalues of ${bf G}$ has analytical form $frac{h}{{{rm{2}}(cos (frac{{kpi }}{{widehat N + 1}}) - 1)}}$ where $k = 1,...,hat{N}$.
Let $hat{mathbf{W}}=begin{pmatrix}
{bf W} & & & & \
& ddots & & & \
& & {bf W} & & \
& & & ddots & \
& & & & {bf W} \
end{pmatrix}$ be a $phat{N}times phat{N}$ diagonal weight matrix repeating $p$ times of $bf {W}$, which is a diagonal weight matrix where all diagonal elements are non-negative and sum to $1$. The weights are free to choose, therefore we can simple choices like every element in $bf W$ is $frac{1}{hat{N}}$
linear-algebra eigenvalues-eigenvectors norm spectral-radius
linear-algebra eigenvalues-eigenvectors norm spectral-radius
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
edited Dec 4 '18 at 15:39
Tony
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
asked Dec 4 '18 at 7:53
TonyTony
1,7561828
1,7561828
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