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Thread: Matrix norms and homotopies.

  1. #1
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    Matrix norms and homotopies.

    I came across a question which I think I know the answer to, but I'd like some confirmation, or a counter-example to show me where my reason fails me (as it often does....darn unreliable brain ).

    the question is this:

    is the subgroup of $\displaystyle \mathrm{GL}_n(\mathbb{R})$, given by:

    $\displaystyle H = \{A \in \mathrm{GL}_n(\mathbb{R})\ |\text{ } \exists \text{ continuous } f:[0,1] \to \mathrm{GL}_n(\mathbb{R}) \text{ with } f(0) = A, f(1) = I \}$

    normal in $\displaystyle \mathrm{GL}_n(\mathbb{R})$?

    the topology used on $\displaystyle \mathrm{GL}_n(\mathbb{R})$ is the standard (metric) topology on $\displaystyle \mathbb{R}^{n^2}$, which I believe induces the frobenius norm on the matrices. this norm is sub-multiplicative, so it seems to me, that if $\displaystyle P \in \mathrm{GL}_n(\mathbb{R})$, that:

    $\displaystyle g:[0,1] \to \mathrm{GL}_n(\mathbb{R})$ given by:

    $\displaystyle g(t) = Pf(t)P^{-1}$ is the desired homotopy of $\displaystyle PAP^{-1}$ with $\displaystyle I$, which would then prove the normality of $\displaystyle H$.

    the fly in the ointment being, that $\displaystyle g$ has to be continuous, which is where the frobenius norm comes in.

    suppose that for $\displaystyle t_0 \in [0,1], f(t_0) = B$, and denote $\displaystyle f(t) = B_t$.

    if $\displaystyle \epsilon > 0$, then if I choose $\displaystyle \delta > 0$ such that:

    $\displaystyle |t - t_0| < \delta \implies |B_t - B| < \frac{\epsilon}{|P||P^{-1}|}$, then

    $\displaystyle |t - t_0| < \delta \implies |g(t) - g(t_0)| = |PB_tP^{-1} - PBP^{-1}| = |P(B_t - B)P^{-1}|$

    $\displaystyle \leq |B_t - B||P||P^{-1}| < \epsilon$, right?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Matrix norms and homotopies.

    Quote Originally Posted by Deveno View Post
    I came across a question which I think I know the answer to, but I'd like some confirmation, or a counter-example to show me where my reason fails me (as it often does....darn unreliable brain ).

    the question is this:

    is the subgroup of $\displaystyle \mathrm{GL}_n(\mathbb{R})$, given by:

    $\displaystyle H = \{A \in \mathrm{GL}_n(\mathbb{R})\ |\text{ } \exists \text{ continuous } f:[0,1] \to \mathrm{GL}_n(\mathbb{R}) \text{ with } f(0) = A, f(1) = I \}$

    normal in $\displaystyle \mathrm{GL}_n(\mathbb{R})$?

    the topology used on $\displaystyle \mathrm{GL}_n(\mathbb{R})$ is the standard (metric) topology on $\displaystyle \mathbb{R}^{n^2}$, which I believe induces the frobenius norm on the matrices. this norm is sub-multiplicative, so it seems to me, that if $\displaystyle P \in \mathrm{GL}_n(\mathbb{R})$, that:

    $\displaystyle g:[0,1] \to \mathrm{GL}_n(\mathbb{R})$ given by:

    $\displaystyle g(t) = Pf(t)P^{-1}$ is the desired homotopy of $\displaystyle PAP^{-1}$ with $\displaystyle I$, which would then prove the normality of $\displaystyle H$.

    the fly in the ointment being, that $\displaystyle g$ has to be continuous, which is where the frobenius norm comes in.

    suppose that for $\displaystyle t_0 \in [0,1], f(t_0) = B$, and denote $\displaystyle f(t) = B_t$.

    if $\displaystyle \epsilon > 0$, then if I choose $\displaystyle \delta > 0$ such that:

    $\displaystyle |t - t_0| < \delta \implies |B_t - B| < \frac{\epsilon}{|P||P^{-1}|}$, then

    $\displaystyle |t - t_0| < \delta \implies |g(t) - g(t_0)| = |PB_tP^{-1} - PBP^{-1}| = |P(B_t - B)P^{-1}|$

    $\displaystyle \leq |B_t - B||P||P^{-1}| < \epsilon$, right?
    Right, this looks fine to me. I think the more general thing, that if someone would have said to you, you would have understood is the conjugation maps are continuous in topological groups. Indeed, suppose that $\displaystyle G$ is a topological group with multiplication map $\displaystyle m:G\times G\to G$ and $\displaystyle i:G\to G$. Then, the map $\displaystyle c_g:G\to G$ given by $\displaystyle h\mapsto ghg^{-1}$ is the composition

    $\displaystyle h\mapsto (h,g) \mapsto (h,g^{-1})\mapsto hg^{-1}\mapsto (g,hg^{-1})\mapsto ghg^{-1}$


    The first map is just the continuous map $\displaystyle G\to G\times\{g\}$, the second is continuous because each coordinate map is continuous, the third is just the continuous multiplication map, the fourth is just the inclusion $\displaystyle G\ \{g\}\times G$, and the last just multiplication again.


    So, if you know that $\displaystyle \text{GL}_n(\mathbb{R})$ is a topological group then you're golden because your function $\displaystyle g$ is just $\displaystyle c_P\circ f$, which is the composition of two continuous functions. But, $\displaystyle \text{GL}_n(\mathbb{R})$ is clearly a topological group since the multiplication maps and inversion maps are just rational functions in each coordinate (in fact, this clearly implies that $\displaystyle \text{GL}_n(\mathbb{R})$ is a Lie group).



    Remark: It's not fruitful to fret over whether a given norm is the one that induces your topology. Indeed, $\displaystyle \text{GL}_n(\mathbb{R})$ is a fin. dim. vector space, and so all norms induce the same topology. But, yes if you fix the usual Euclidean norm for $\displaystyle \mathbb{R}^{n^2}$ then this norm is carried naturally by the obvious identification $\displaystyle \mathbb{R}^{n^2}\approx\text{GL}_n(\mathbb{R})$ to the Frobenius norm.
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    Re: Matrix norms and homotopies.

    So the question is, is the connected component containing I, as a subgroup of $\displaystyle G=GL_n(R)$, normal or not?
    Note that $\displaystyle GL_n(R)$ has only two components, H={det>0} and K={det<0}, where det is the determinant function.
    A subgroup H of G is said to be normal if Hg=gH for any g in G. If $\displaystyle g \in H$, we have gH=Hg=H certainly.
    If g is not in H, det(g)<0, we have det(gh)=det(g)det(h)<0, that is, $\displaystyle gh \in K$, so gH is contained in K.
    And for any $\displaystyle a \in K, a=g*(g^{-1}a)$, and $\displaystyle det(g^{-1}a)=det(a)/det(g)>0$, so $\displaystyle g^{-1}a \in H$,
    So $\displaystyle a \in gH$, that is K is contained in gH. So we have gH=K. Similarly we have Hg=K. So gH=Hg. H is normal.
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    Re: Matrix norms and homotopies.

    Quote Originally Posted by Drexel28 View Post
    Right, this looks fine to me. I think the more general thing, that if someone would have said to you, you would have understood is the conjugation maps are continuous in topological groups. Indeed, suppose that $\displaystyle G$ is a topological group with multiplication map $\displaystyle m:G\times G\to G$ and $\displaystyle i:G\to G$. Then, the map $\displaystyle c_g:G\to G$ given by $\displaystyle h\mapsto ghg^{-1}$ is the composition

    $\displaystyle h\mapsto (h,g) \mapsto (h,g^{-1})\mapsto hg^{-1}\mapsto (g,hg^{-1})\mapsto ghg^{-1}$


    The first map is just the continuous map $\displaystyle G\to G\times\{g\}$, the second is continuous because each coordinate map is continuous, the third is just the continuous multiplication map, the fourth is just the inclusion $\displaystyle G\ \{g\}\times G$, and the last just multiplication again.


    So, if you know that $\displaystyle \text{GL}_n(\mathbb{R})$ is a topological group then you're golden because your function $\displaystyle g$ is just $\displaystyle c_P\circ f$, which is the composition of two continuous functions. But, $\displaystyle \text{GL}_n(\mathbb{R})$ is clearly a topological group since the multiplication maps and inversion maps are just rational functions in each coordinate (in fact, this clearly implies that $\displaystyle \text{GL}_n(\mathbb{R})$ is a Lie group).
    d'oh! conjugation is continuous, right. should i start dyeing my hair blonde?



    Remark: It's not fruitful to fret over whether a given norm is the one that induces your topology. Indeed, $\displaystyle \text{GL}_n(\mathbb{R})$ is a fin. dim. vector space, and so all norms induce the same topology. But, yes if you fix the usual Euclidean norm for $\displaystyle \mathbb{R}^{n^2}$ then this norm is carried naturally by the obvious identification $\displaystyle \mathbb{R}^{n^2}\approx\text{GL}_n(\mathbb{R})$ to the Frobenius norm.
    well, when i first came across the problem, the usual topology (for $\displaystyle \text{GL}_n(\mathbb{R}))$ was indicated, so it seemed appropriate to use the usual norm for the vectorization of a matrix. since i was trying to prove the continuity of g directly, i needed the sub-multiplicative property, and not all norms are sub-multiplicative (although that can be fixed by re-scaling).

    Quote Originally Posted by xxp9 View Post
    So the question is, is the connected component containing I, as a subgroup of $\displaystyle G=GL_n(R)$, normal or not?
    Note that $\displaystyle GL_n(R)$ has only two components, H={det>0} and K={det<0}, where det is the determinant function.
    A subgroup H of G is said to be normal if Hg=gH for any g in G. If $\displaystyle g \in H$, we have gH=Hg=H certainly.
    If g is not in H, det(g)<0, we have det(gh)=det(g)det(h)<0, that is, $\displaystyle gh \in K$, so gH is contained in K.
    And for any $\displaystyle a \in K, a=g*(g^{-1}a)$, and $\displaystyle det(g^{-1}a)=det(a)/det(g)>0$, so $\displaystyle g^{-1}a \in H$,
    So $\displaystyle a \in gH$, that is K is contained in gH. So we have gH=K. Similarly we have Hg=K. So gH=Hg. H is normal.
    great answer. i wasn't thinking in terms of homotopies being paths in $\displaystyle \text{GL}_n(\mathbb{R})$, which makes it rather simple, because det is continuous.
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