Can we prove that every neighborhood of the $0$ companion matrix that contains a Hurwitz matrix?
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Let $mathcal C subset M_n(mathbb R)$ be the set of companion matrices which we can identify by $mathbb R^n$. Let us denote the set of companion matrices with eigenvalues lying on the open left half plane by $mathcal H$, i.e.,
begin{align*}
mathcal H = {A in M_n(mathbb R): A text{ is in companion form and } max_i text{Re}(lambda_i(A)) < 0}.
end{align*}
Let $A$ be
begin{align*}
A=begin{pmatrix} 0 & cdots & 0& 0 \ 1 & cdots & 0 & 0\ vdots &ddots & vdots &vdots \ 0 &cdots & 1 & 0 end{pmatrix}
end{align*}
Is it true that for every neighborhood $B_{varepsilon}$ of $A$ in $mathcal C$ we have $B_{varepsilon} cap mathcal H neq emptyset$? In other words, $A in partial mathcal H$.
The statement is obviously true for $n=1, 2, 3, 4$. This is due to Routh-Hurwitz condition. We only need to choose appropriate coefficients. But as the dimension goes up, this condition becomes unmanageable. In some sense, I feel the general case should also be true but could not prove it.
linear-algebra general-topology matrix-analysis
asked 2 days ago
user9527
1,2921627
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up vote
1
down vote
favorite
Let $mathcal C subset M_n(mathbb R)$ be the set of companion matrices which we can identify by $mathbb R^n$. Let us denote the set of companion matrices with eigenvalues lying on the open left half plane by $mathcal H$, i.e.,
begin{align*}
mathcal H = {A in M_n(mathbb R): A text{ is in companion form and } max_i text{Re}(lambda_i(A)) < 0}.
end{align*}
Let $A$ be
begin{align*}
A=begin{pmatrix} 0 & cdots & 0& 0 \ 1 & cdots & 0 & 0\ vdots &ddots & vdots &vdots \ 0 &cdots & 1 & 0 end{pmatrix}
end{align*}
Is it true that for every neighborhood $B_{varepsilon}$ of $A$ in $mathcal C$ we have $B_{varepsilon} cap mathcal H neq emptyset$? In other words, $A in partial mathcal H$.
The statement is obviously true for $n=1, 2, 3, 4$. This is due to Routh-Hurwitz condition. We only need to choose appropriate coefficients. But as the dimension goes up, this condition becomes unmanageable. In some sense, I feel the general case should also be true but could not prove it.
linear-algebra general-topology matrix-analysis
asked 2 days ago
user9527
1,2921627
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $mathcal C subset M_n(mathbb R)$ be the set of companion matrices which we can identify by $mathbb R^n$. Let us denote the set of companion matrices with eigenvalues lying on the open left half plane by $mathcal H$, i.e.,
begin{align*}
mathcal H = {A in M_n(mathbb R): A text{ is in companion form and } max_i text{Re}(lambda_i(A)) < 0}.
end{align*}
Let $A$ be
begin{align*}
A=begin{pmatrix} 0 & cdots & 0& 0 \ 1 & cdots & 0 & 0\ vdots &ddots & vdots &vdots \ 0 &cdots & 1 & 0 end{pmatrix}
end{align*}
Is it true that for every neighborhood $B_{varepsilon}$ of $A$ in $mathcal C$ we have $B_{varepsilon} cap mathcal H neq emptyset$? In other words, $A in partial mathcal H$.
The statement is obviously true for $n=1, 2, 3, 4$. This is due to Routh-Hurwitz condition. We only need to choose appropriate coefficients. But as the dimension goes up, this condition becomes unmanageable. In some sense, I feel the general case should also be true but could not prove it.
linear-algebra general-topology matrix-analysis
asked 2 days ago
user9527
1,2921627
Let $mathcal C subset M_n(mathbb R)$ be the set of companion matrices which we can identify by $mathbb R^n$. Let us denote the set of companion matrices with eigenvalues lying on the open left half plane by $mathcal H$, i.e.,
begin{align*}
mathcal H = {A in M_n(mathbb R): A text{ is in companion form and } max_i text{Re}(lambda_i(A)) < 0}.
end{align*}
Let $A$ be
begin{align*}
A=begin{pmatrix} 0 & cdots & 0& 0 \ 1 & cdots & 0 & 0\ vdots &ddots & vdots &vdots \ 0 &cdots & 1 & 0 end{pmatrix}
end{align*}
Is it true that for every neighborhood $B_{varepsilon}$ of $A$ in $mathcal C$ we have $B_{varepsilon} cap mathcal H neq emptyset$? In other words, $A in partial mathcal H$.
The statement is obviously true for $n=1, 2, 3, 4$. This is due to Routh-Hurwitz condition. We only need to choose appropriate coefficients. But as the dimension goes up, this condition becomes unmanageable. In some sense, I feel the general case should also be true but could not prove it.
linear-algebra general-topology matrix-analysis
linear-algebra general-topology matrix-analysis
asked 2 days ago
user9527
1,2921627
asked 2 days ago
user9527
1,2921627
edited 2 days ago
asked 2 days ago
user9527
1,2921627
asked 2 days ago
user9527
1,2921627
asked 2 days ago
user9527
1,2921627
1,2921627
add a comment |
add a comment |
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