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Proofs about invertible linear functions

  1. Mar 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Let [itex]G\subset L(\mathbb{R}^n;\mathbb{R}^n)[/itex] be the subset of invertible linear transformations.

    a) For [itex]H\in L(\mathbb{R}^n;\mathbb{R}^n)[/itex], prove that if [itex]||H||<1[/itex], then the partial sum [itex]L_n=\sum_{k=0}^{n}H^k[/itex] converges to a limit [itex]L[/itex] and [itex]||L||\leq\frac{1}{1-||H||}[/itex].

    b) If [itex]A\in L(\mathbb{R}^n;\mathbb{R}^n)[/itex] satisfies [itex]||A-I||<1[/itex], then [itex]A[/itex] is invertible and [itex]A^{-1}=\sum_{k=0}^{\infty }H^k[/itex] where [itex]I-A=H[/itex]. (Hint: Show that [itex]AL_n=H^{n+1}[/itex])

    c) Let [itex]\varphi :G\rightarrow G[/itex] be the inversion map [itex]\varphi(A)=A^{-1}[/itex]. Prove that [itex]\varphi[/itex] is continuous at the identity [itex]I[/itex], using the previous two facts.

    d) Let [itex]A, C \in G[/itex] and [itex]B=A^{-1}[/itex]. We can write [itex]C=A-K[/itex] and [itex]\varphi(A-K)=c^{-1}=A^{-1}(I-H)^{-1}[/itex] where [itex]H=BK[/itex]. Use this to prove that [itex]\varphi[/itex] is continuous at [itex]A[/itex].






    2. Relevant equations

    n/a

    3. The attempt at a solution
    a) is easy by using the geometric series. For c and d, what [itex]\varphi[/itex] is continuous at the identity [itex]I[/itex] and [itex]\varphi[/itex] is continuous at [itex]A[/itex] mean? What we need to prove?
     
    Last edited: Mar 15, 2013
  2. jcsd
  3. Mar 15, 2013 #2

    micromass

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    Hint: Absolute convergence.
     
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