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Diagonalisability problem and others

  1. Feb 22, 2010 #1
    1. The problem statement, all variables and given/known data
    1.Prove that if A is a real matrix then At A is diagonalisable.

    2. Given a known 3*3 matrix A, Calculate the maximum and minimum values of ||Ax|| on the sphere ||x|| = 1.

    2. Relevant equations

    3. The attempt at a solution
    For the first problem, I'm thinking of proving that AtA is symmetric, but I'm not sure which properties to use.
    For the second one, is X a 3*n matrix? Do I need to discuss n?
    Any help is greatly appreciated!
  2. jcsd
  3. Feb 22, 2010 #2


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    If [itex]B=A^TA[/itex], write down what [itex]B_{ij}[/itex] and [itex]B_{ji}[/itex] are in terms of the elements of A and show that they're equal.
  4. Feb 23, 2010 #3
    Thanks a lot! I can't believe I missed it in the first place.
    Could anyone give me some hints about problem 2?
  5. Feb 23, 2010 #4


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    For the second one, yes, x must be such a vector that the product Ax is meaningful, i.e. a 3x1 vector (matrix).
  6. Feb 23, 2010 #5
    thx, I finished it!
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