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Paths within GL^+(n,R)

  1. Aug 11, 2014 #1

    WWGD

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    Hi all, I am trying to construct a continuous matrix-valued function h=h(f,g) ; f,g in C^1[a,b],all defined in an open interval (a,b) on the Real line with some given properties, or at least I want to show that the function exists . I have/know that for a,b in [a,b], both Det(h(a))>0 and Det(h(b))>0 . Then I want, for all t in (a,b), that the determinant of :

    [tex] A=
    \begin{pmatrix}
    f(t) & g(t) \\
    f '(t)& g'(t) \\

    \end{pmatrix}
    [/tex]

    will be positive . I am trying to use that GL(n,R) has two connected components; GL^+ and
    Gl^- ;matrices with positive and negative determinant respectively (e.g., use the fact that
    Gaussian elimination can take any A in GL^+ into the Id. ).

    Then I think can argue that since f,g ;f',g' are continuous by assumption, the map h(f,g) is
    itself also continuous , so that h[a,b] -- remember here that [a,b] is an interval in the real line--is connected, and so h[a,b] must then lie in GL^+(n,R), for all t in [a,b], i.e., Det(h[a,b])>0.
    Is this right?
     
  2. jcsd
  3. Aug 12, 2014 #2
    Consider the function [tex]h:C^1([a,b])\times C^1([a,b])\to GL_1^+(\mathbb{R})\cong\mathbb{R}_+,~(f,g)\mapsto\left\{\begin{matrix}2 & \text{if } f(0)+g(0)\in\mathbb{Q} \\ 1 & \text{if } f(0)+g(0)\not\in\mathbb{Q}\end{matrix}\right.~.[/tex]

    The determinant of the [itex]1\times 1[/itex] matrix [itex]h(f,g)[/itex] is always greater than 0, and the function is nowhere continuous.

    Why is [itex]A[/itex] important in this? It seems unrelated.
     
  4. Aug 13, 2014 #3

    WWGD

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    Thanks, what I am actually trying to do is to start with any matrix A with DetA>0 and show it can be continuously-deformed into any other matrix B with DetB>0 , and the deformation is done within GL^+(2,R), i.e., I am trying to show GL^+(2,R) , as a Lie group (with the subspace topology of R^4)is path-connected. I think using elementary row operations--seen as continuous maps-- the right way , transforming A into the Id is /gives us the path between A and the Id, showing path-connectedness . Still, we need to find row operations that preserve the sign of Det.
    I am trying to construct a collection of continuous functions parametrized by an interval , taking A to Id, i.e., a path between A and Id., to show GL^+(2,R) is path-connected (obviously this holds for GL(n,R) for all natural n) .


    Sorry for being unclear; I am typing with an OSK.
     
    Last edited: Aug 13, 2014
  5. Aug 13, 2014 #4
    When in doubt, check Stack Exchange.
     
    Last edited: Aug 13, 2014
  6. Aug 14, 2014 #5

    WWGD

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    Thanks, Dragon; and they also proved path-connectedness, good ref.
     
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