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Unitary coordinate transformation = rotation?

  1. Jun 8, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose I define a linear coordinate transformation that I can describe with a matrix U.
    If U is unitary. i.e.
    [tex]U^{-1}U = UU^{-1}=1[/tex]
    does that necessarily imply that the transformation corresponds to a pure rotation (plus maybe a translation), so that I may assume that volumes are invariant?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jun 8, 2008 #2


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    Science Advisor
    Homework Helper

    Yes, volumes are invariant, certainly. The jacobian is 1. There are no translations if the transformation is linear. There could be reflections.
  4. Jun 9, 2008 #3
    Thank you.
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