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Ortogonal matrix

  1. Jan 16, 2008 #1
    [SOLVED] Ortogonal matrix

    1. The problem statement, all variables and given/known data
    I have a 2x2 matrix A:

    [alfa beta]
    [beta -alfa],

    where alfa and beta are real parameters. I have to find out for which values of alfa and beta A is an orthogonal matrix.

    3. The attempt at a solution
    A matrix is orthogonal if it satisfies Q*Q^T = I.

    So I will multiply A with A^T and equal it to I, and I get the condition alfa^2 + beta^2 = 1. Are there any other conditions I need?
     
  2. jcsd
  3. Jan 16, 2008 #2
    ok.. so u have a matrix:

    [tex]
    \begin{bmatrix}
    \alpha & \beta \\
    \beta & -\alpha
    \end{bmatrix}
    [/tex]

    on multiplying with it's transpose, you have:

    [tex]
    \begin{bmatrix}
    \alpha^2 + \beta^2 & 0\\
    0 & \beta^2 - \alpha^2
    \end{bmatrix} =
    \begin{bmatrix}
    1 & 0\\
    0 & 1
    \end{bmatrix}
    [/tex]

    Look at the matrix now, equated with the identity matrix. You've taken one equation correctly. But, does the relation we have now provide you with another equation?

    HINT: For two matrices to be equal, all their elements should be equal.
     
  4. Jan 16, 2008 #3
    Hmm, when multiplying with it's inverse (transpose), I get:


    [a^2+b^2 0 ]
    [ 0 a^2+b^2].
     
    Last edited: Jan 16, 2008
  5. Jan 16, 2008 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, you were right the first time. Given only the information that A is a 2x2 orthogonal matrix you only have [itex]\alpha^2+ \beta^2= 1[/itex]. That's because there are an infinite number of such matrices, not just one. Any [itex]\alpha[/itex] and [itex]\beta[/itex] satisfying [itex]\alpha^2+ \beta^2= 1[/itex] will give you an orthogonal matrix.
     
  6. Jan 16, 2008 #5
    Cool, thanks.
     
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