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How many connected components could the intersection of ${A in M_n(mathbb R): rho(A) < 1}$ and an affine...
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Let $mathcal E = {A in M_n(mathbb R): rho(A) < 1}$ where $rho(cdot)$ is the spectral radius and $mathcal U$ be an affine space in $M_n(mathbb R)$. If we assume $mathcal E cap mathcal U neq emptyset$, how many connected components could the intersection have?
In proving $mathcal E$ is connected, I know we can use a path $(1-t)A + t 0$ but if $B$ is in the intersection, $(1-t)B$ could not be guaranteed in the intersection.
linear-algebra general-topology path-connected
asked Nov 15 at 20:54
user9527
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up vote
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down vote
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Let $mathcal E = {A in M_n(mathbb R): rho(A) < 1}$ where $rho(cdot)$ is the spectral radius and $mathcal U$ be an affine space in $M_n(mathbb R)$. If we assume $mathcal E cap mathcal U neq emptyset$, how many connected components could the intersection have?
In proving $mathcal E$ is connected, I know we can use a path $(1-t)A + t 0$ but if $B$ is in the intersection, $(1-t)B$ could not be guaranteed in the intersection.
linear-algebra general-topology path-connected
asked Nov 15 at 20:54
user9527
1,2041627
This question has an open bounty worth +100
reputation from user9527 ending in 7 days.
This question has not received enough attention.
Intersections of convex sets are convex
– Bananach
Nov 15 at 21:23
3
$mathcal E$ is not convex.
– user9527
Nov 15 at 21:29
Just as a curious observer could you provide an example of two matrices whose path leaves $mathcal{E}$?
– user25959
Nov 15 at 21:32
1
@user25959 You can take $pmatrix{a & 10\ 0 & a}$ and $pmatrix{b & 0 \10 & b}$. Make suitable choices of $a, b$.
– user9527
Nov 15 at 22:05
1
@user25959 he is right, and $a=b=0$ already suffice to show it. I got confused with the spectral radius vs. operator norm
– Bananach
Nov 16 at 6:16
|
show 2 more comments
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Let $mathcal E = {A in M_n(mathbb R): rho(A) < 1}$ where $rho(cdot)$ is the spectral radius and $mathcal U$ be an affine space in $M_n(mathbb R)$. If we assume $mathcal E cap mathcal U neq emptyset$, how many connected components could the intersection have?
In proving $mathcal E$ is connected, I know we can use a path $(1-t)A + t 0$ but if $B$ is in the intersection, $(1-t)B$ could not be guaranteed in the intersection.
linear-algebra general-topology path-connected
asked Nov 15 at 20:54
user9527
1,2041627
Let $mathcal E = {A in M_n(mathbb R): rho(A) < 1}$ where $rho(cdot)$ is the spectral radius and $mathcal U$ be an affine space in $M_n(mathbb R)$. If we assume $mathcal E cap mathcal U neq emptyset$, how many connected components could the intersection have?
In proving $mathcal E$ is connected, I know we can use a path $(1-t)A + t 0$ but if $B$ is in the intersection, $(1-t)B$ could not be guaranteed in the intersection.
linear-algebra general-topology path-connected
linear-algebra general-topology path-connected
asked Nov 15 at 20:54
user9527
1,2041627
asked Nov 15 at 20:54
user9527
1,2041627
asked Nov 15 at 20:54
user9527
1,2041627
asked Nov 15 at 20:54
user9527
1,2041627
asked Nov 15 at 20:54
user9527
1,2041627
1,2041627
This question has an open bounty worth +100
reputation from user9527 ending in 7 days.
This question has not received enough attention.
This question has an open bounty worth +100
reputation from user9527 ending in 7 days.
This question has not received enough attention.
Intersections of convex sets are convex
– Bananach
Nov 15 at 21:23
3
$mathcal E$ is not convex.
– user9527
Nov 15 at 21:29
Just as a curious observer could you provide an example of two matrices whose path leaves $mathcal{E}$?
– user25959
Nov 15 at 21:32
1
@user25959 You can take $pmatrix{a & 10\ 0 & a}$ and $pmatrix{b & 0 \10 & b}$. Make suitable choices of $a, b$.
– user9527
Nov 15 at 22:05
1
@user25959 he is right, and $a=b=0$ already suffice to show it. I got confused with the spectral radius vs. operator norm
– Bananach
Nov 16 at 6:16
|
show 2 more comments
Intersections of convex sets are convex
– Bananach
Nov 15 at 21:23
3
$mathcal E$ is not convex.
– user9527
Nov 15 at 21:29
Just as a curious observer could you provide an example of two matrices whose path leaves $mathcal{E}$?
– user25959
Nov 15 at 21:32
1
@user25959 You can take $pmatrix{a & 10\ 0 & a}$ and $pmatrix{b & 0 \10 & b}$. Make suitable choices of $a, b$.
– user9527
Nov 15 at 22:05
1
@user25959 he is right, and $a=b=0$ already suffice to show it. I got confused with the spectral radius vs. operator norm
– Bananach
Nov 16 at 6:16
Intersections of convex sets are convex
– Bananach
Nov 15 at 21:23
Intersections of convex sets are convex
– Bananach
Nov 15 at 21:23
3
3
$mathcal E$ is not convex.
– user9527
Nov 15 at 21:29
$mathcal E$ is not convex.
– user9527
Nov 15 at 21:29
Just as a curious observer could you provide an example of two matrices whose path leaves $mathcal{E}$?
– user25959
Nov 15 at 21:32
Just as a curious observer could you provide an example of two matrices whose path leaves $mathcal{E}$?
– user25959
Nov 15 at 21:32
1
1
@user25959 You can take $pmatrix{a & 10\ 0 & a}$ and $pmatrix{b & 0 \10 & b}$. Make suitable choices of $a, b$.
– user9527
Nov 15 at 22:05
@user25959 You can take $pmatrix{a & 10\ 0 & a}$ and $pmatrix{b & 0 \10 & b}$. Make suitable choices of $a, b$.
– user9527
Nov 15 at 22:05
1
1
@user25959 he is right, and $a=b=0$ already suffice to show it. I got confused with the spectral radius vs. operator norm
– Bananach
Nov 16 at 6:16
@user25959 he is right, and $a=b=0$ already suffice to show it. I got confused with the spectral radius vs. operator norm
– Bananach
Nov 16 at 6:16
|
show 2 more comments
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Intersections of convex sets are convex
– Bananach
Nov 15 at 21:23
3
$mathcal E$ is not convex.
– user9527
Nov 15 at 21:29
Just as a curious observer could you provide an example of two matrices whose path leaves $mathcal{E}$?
– user25959
Nov 15 at 21:32
1
@user25959 You can take $pmatrix{a & 10\ 0 & a}$ and $pmatrix{b & 0 \10 & b}$. Make suitable choices of $a, b$.
– user9527
Nov 15 at 22:05
1
@user25959 he is right, and $a=b=0$ already suffice to show it. I got confused with the spectral radius vs. operator norm
– Bananach
Nov 16 at 6:16