Results 1 to 3 of 3

Math Help - Eigenvalue, norm

  1. #1
    Senior Member
    Joined
    Nov 2008
    Posts
    461

    Eigenvalue, norm

    Hello!

    I've got a question according the solution of this exercise

    A \in M(n, \mathbb{K})

    Show that  \|A \| := \sqrt{E_{max}}, where

    E_{max} is the biggest eigenvalue of A^*A

    and

    \| A \| := \sup_{\| x \|_2 = 1} \| Ax\|^2 = \sup_{\| x\| \le 1} \langle Ax, Ax \rangle

    = \sup \langle A^* Ax,x \rangle

    SOLUTION

    A^*A is positive definit, so there is a unitary U \in GL(n, \mathbb{K}) :


    " alt="UA^*AU^{-1} = diag (E_1, ... , E_n )=" /> and E_i \ge 0 denote the eigenvalues of A^*A


    So


    \| A \|^2 = \sup \langle A^*Ax, x \rangle

    = \sup \langle U^{-1}DUx,x \rangle

    = \sup \langle DUx , Ux \rangle , \ Ux =: y

    =\sup \langle Dy, y \rangle

    = \sup \sum^n_{i=1} E_i |y_i |^2

    = E_{max}

    So why is the last line? Why is it not \ge E_{max}?


    Rapha
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    That is because y = Ux is supposed to be a unit vector, so \sum^n_{i=1} |y_i |^2 = 1. Therefore \sum^n_{i=1} E_i |y_i |^2\leqslant \sum^n_{i=1} E_{\max} |y_i |^2\leqslant E_{\max}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2008
    Posts
    461
    Hello Opalg!

    Thanks alot.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lp-norm converges to Chebyshev norm?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 12th 2010, 09:53 AM
  2. Replies: 3
    Last Post: July 13th 2010, 07:37 PM
  3. Replies: 2
    Last Post: November 7th 2009, 01:13 PM
  4. Euclidean Norm and Maximum Norm
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 7th 2009, 05:26 AM
  5. Vector Norm and Matrix Norm
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 18th 2008, 11:49 AM

Search Tags


/mathhelpforum @mathhelpforum