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







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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















up vote
4
down vote

favorite
1












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







share|cite|improve this question















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













up vote
4
down vote

favorite
1









up vote
4
down vote

favorite
1






1





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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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


















  • 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















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