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Homework Help: Linear Algebra- find an orthogonal matrix with eigenvalue!=1 or -1

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data
    I have to find an orthogonal matrix with an eigenvalue that does not equal 1 or -1. That's it. I'm completely stumped.

    2. Relevant equations
    An orthogonal matrix is defined as a matrix whose columns are an orthonormal basis, that is they are all orthogonal to each other and each vector has length 1. These matrices have the property that their inverse is the same as their transpose. I don't think their are any other equations.

    3. The attempt at a solution
    My professor claims that this is possible. So far I thought about a 1x1 matrix, as that is defined as each vector being orthogonal to each other, but the vector only has length 1 if the matrix is [1] or [-1]. And rectangular matrices don't have inverses. I'm stumped.
     
  2. jcsd
  3. Apr 14, 2010 #2

    Mark44

    Staff: Mentor

    See if you can cook up a 2x2 matrix that is orthogonal and whose eigenvalues are neither 1 nor -1. Don't limit yourself to real eigenvalues.
     
  4. Apr 14, 2010 #3

    Mark44

    Staff: Mentor

    BTW, this really should be in the Calculus & Beyond section.
     
  5. Apr 14, 2010 #4
    Thanks for the help. I found one:
    1/sqrt2 -1/sqrt2
    1/sqrt2 1/sqrt2

    It's orthogonal and has nonreal eigenvalues. Sorry about putting this post in the wrong forum. I always thought of linear algebra as being a "lower" math than calculus.
     
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