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Condition on real matrix $A$ such that the set $S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0 $ is connected
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This is closed related to the question I asked here which concerning the number of connected components in $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$. Although it has not received an answer, but I suspect the answer will be dependent on $A$. Here I will put some condition on $A$ and be interested in sufficient conditions guaranteeing $E$ is connected.
Let $A in M_n(mathbb R)$ be such $Ae_1 = e_2$ where $e_1, e_2$ are standard basis in $mathbb R^n$. Let $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$, i.e., the conjugacy class of $A$ but with restriction that first column of $S^-1AS$ to be $e_2$. Now I am interested in sufficient conditions on $A$ such that $E$ has only $1$ connected components.
Here are some related questions:
Connectedness of matrix conjugacy class,
connectedness of matrix conjugacy classes of a fixed real $A$ but with the first column of $A$ invariant.
Both questions haven't received answers at this moment, but some of the comments might be useful.
linear-algebra abstract-algebra general-topology connectedness path-connected
asked Jul 21 at 1:08
user9527
925525
add a comment |Â
up vote
1
down vote
favorite
This is closed related to the question I asked here which concerning the number of connected components in $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$. Although it has not received an answer, but I suspect the answer will be dependent on $A$. Here I will put some condition on $A$ and be interested in sufficient conditions guaranteeing $E$ is connected.
Let $A in M_n(mathbb R)$ be such $Ae_1 = e_2$ where $e_1, e_2$ are standard basis in $mathbb R^n$. Let $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$, i.e., the conjugacy class of $A$ but with restriction that first column of $S^-1AS$ to be $e_2$. Now I am interested in sufficient conditions on $A$ such that $E$ has only $1$ connected components.
Here are some related questions:
Connectedness of matrix conjugacy class,
connectedness of matrix conjugacy classes of a fixed real $A$ but with the first column of $A$ invariant.
Both questions haven't received answers at this moment, but some of the comments might be useful.
linear-algebra abstract-algebra general-topology connectedness path-connected
asked Jul 21 at 1:08
user9527
925525
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
This is closed related to the question I asked here which concerning the number of connected components in $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$. Although it has not received an answer, but I suspect the answer will be dependent on $A$. Here I will put some condition on $A$ and be interested in sufficient conditions guaranteeing $E$ is connected.
Let $A in M_n(mathbb R)$ be such $Ae_1 = e_2$ where $e_1, e_2$ are standard basis in $mathbb R^n$. Let $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$, i.e., the conjugacy class of $A$ but with restriction that first column of $S^-1AS$ to be $e_2$. Now I am interested in sufficient conditions on $A$ such that $E$ has only $1$ connected components.
Here are some related questions:
Connectedness of matrix conjugacy class,
connectedness of matrix conjugacy classes of a fixed real $A$ but with the first column of $A$ invariant.
Both questions haven't received answers at this moment, but some of the comments might be useful.
linear-algebra abstract-algebra general-topology connectedness path-connected
asked Jul 21 at 1:08
user9527
925525
This is closed related to the question I asked here which concerning the number of connected components in $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$. Although it has not received an answer, but I suspect the answer will be dependent on $A$. Here I will put some condition on $A$ and be interested in sufficient conditions guaranteeing $E$ is connected.
Let $A in M_n(mathbb R)$ be such $Ae_1 = e_2$ where $e_1, e_2$ are standard basis in $mathbb R^n$. Let $E := S^-1AS: S in GL_n(mathbb R), (SA-AS)e_1 = 0$, i.e., the conjugacy class of $A$ but with restriction that first column of $S^-1AS$ to be $e_2$. Now I am interested in sufficient conditions on $A$ such that $E$ has only $1$ connected components.
Here are some related questions:
Connectedness of matrix conjugacy class,
connectedness of matrix conjugacy classes of a fixed real $A$ but with the first column of $A$ invariant.
Both questions haven't received answers at this moment, but some of the comments might be useful.
linear-algebra abstract-algebra general-topology connectedness path-connected
asked Jul 21 at 1:08
user9527
925525
asked Jul 21 at 1:08
user9527
925525
asked Jul 21 at 1:08
user9527
925525
asked Jul 21 at 1:08
user9527
925525
925525
add a comment |Â
add a comment |Â
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