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Linear Algebra I: Orthogonal Matrix Condition

  1. Dec 10, 2011 #1

    Dembadon

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    Gold Member

    I would like to check my reasoning for this problem to make sure I understand what an orthogonal matrix is.

    1. The problem statement, all variables and given/known data

    Determine if the matrix is orthogonal. If orthogonal, find the inverse.

    [tex]
    \begin{pmatrix}
    -1 & 2 & 2\\
    2 & -1 & 2\\
    2 & 2 & -1
    \end{pmatrix}
    [/tex]

    2. Relevant equations

    If a matrix [itex]A[/itex] is orthogonal, then
    [tex]
    A^{-1} = A^T.
    [/tex]

    3. The attempt at a solution

    One of the conditions that must be met for a matrix to be orthogonal is that the length of the vectors spanning its column space must be 1, correct? So, if we let
    [tex]
    A=\begin{pmatrix}
    -1 & 2 & 2\\
    2 & -1 & 2\\
    2 & 2 & -1
    \end{pmatrix}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3),
    [/tex]
    then
    [tex]
    ||\mathbf{a}_1||^2 \neq 1,
    [/tex]
    so a condition for orthogonality has been violated. Thus, [itex]A[/itex] is not orthogonal and there is no need to continue with the problem.

    Another way to put it would be to say that the [itex]Col\ A[/itex] is not an orthonormal set, so [itex]A[/itex] is not orthogonal. Is this correct?
     
  2. jcsd
  3. Dec 10, 2011 #2

    micromass

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    Staff Emeritus
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    2016 Award

    That is all correct!!
     
  4. Dec 10, 2011 #3

    Dembadon

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    Gold Member

    Thanks, micro! :smile:
     
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