Results 1 to 2 of 2

Math Help - Matrix Norms & Normal Matrix Problem

  1. #1
    Junior Member
    Joined
    Jun 2010
    Posts
    54

    Matrix Norms & Normal Matrix Problem

    I would like some direction on this problem please:

    Let A be a complex n x n matrix, with ||A||2 = ||A2||. Prove that A is normal.

    I believe the norm being used is the norm which takes the largest value of |Ax| over all vectors x of length 1. Unfortunately, and quite annoyingly, the problem set doesn't specify. However, there was another problem which used an unspecified norm, and this norm seemed to be the one which made sense, so I believe this norm is the one being used.

    Here's my idea: first assume that A is upper-triangular. Prove that A must be diagonal. Then use Schur's theorem (a matrix can be unitarily upper triangularized) to prove the general case. But the reduction to upper triangular matrices hasn't yet helped me very much.

    Another idea is to show that x*AA*x = x*A*Ax for all x, from which the normality of A follows. A possibly-relevant fact is that ||A|| is the largest singular value of A. Perhaps I can use the factorization A=\sqrt{A^*A}\ U for some unitary U. Obviously, none of these have yet led me to a solution. Any ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jun 2010
    Posts
    54

    Re: Matrix Norms & Normal Matrix Problem

    MY APOLOGIES!

    The question specifies that A is a linear map on a TWO-DIMENSIONAL complex Euclidean space. I FAIL.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vector Matrix vs Matrix Norms
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 29th 2012, 10:42 AM
  2. Matrix norms and homotopies.
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: December 1st 2011, 02:01 PM
  3. Matrix Norms
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: September 18th 2010, 10:07 PM
  4. vector norms and matrix norms
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 13th 2010, 02:42 PM
  5. matrix norms
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 15th 2009, 11:44 AM

Search Tags


/mathhelpforum @mathhelpforum