1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Co-norm of an invertible linear transformation on R^n

  1. Mar 15, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex]|\;|[/itex] is a norm on [itex]\mathbb{R}^n[/itex].
    Define the co-norm of the linear transformation [itex]T : \mathbb{R}^n\rightarrow\mathbb{R}^n[/itex] to be
    [itex]m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \}[/itex]
    Prove that if [itex]T[/itex] is invertible with inverse [itex]S[/itex] then [itex]m(T)=\frac{1}{||S||}[/itex].


    2. Relevant equations
    n/a


    3. The attempt at a solution
    I think probably we need to do something with the norm, but I still can't get it... So thank you.
     
  2. jcsd
  3. Mar 15, 2013 #2

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Equalities of the form inf X = A are often proved by showing that the two inequalities inf X ≤ A and inf X ≥ A both hold. Together they imply equality of course. One of these proofs will typically use that inf X a lower bound of X (consider an arbitrary member of X), and the other will typically use that inf is the greatest lower bound of the set.

    How is ##\|S\|## defined? Can you prove anything about the relationship between ##\|S\|## and ##\|T\|##?

    Edit: I have so far only proved the inequality ##m(T)\leq 1/\|S\|##. The idea that I think looks the most promising for a proof of the equality is to take a closer look at the set ##\{|Tx|:|x|=1\}##. What is its infimum? What is its supremum?
     
    Last edited: Mar 15, 2013
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Co-norm of an invertible linear transformation on R^n
Loading...