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Schur decomposition ambiguity

  1. Apr 22, 2009 #1
    Wikipedia defines the Shur decomposition of matrix A as

    A = Q U Q^{-1}

    where Q is unitary and U is upper triangular.

    http://en.wikipedia.org/wiki/Schur_decomposition

    Mathworld defines the Shur decomposition of matrix A as

    Q^H A Q = T,

    where Q is unitary and T is upper triangular.

    http://mathworld.wolfram.com/SchurDecomposition.html

    Because Q is unitary, the inverse is the same as the conjugate transpose...but they still seem like completely different definitions because the matrix is either on the inside or the outside. What's the truth?
     
  2. jcsd
  3. Apr 23, 2009 #2
    Since Q^H = Q^-1, you have

    Q Q^H A Q = Q T
    =>
    Q Q^-1 A Q = Q T
    =>
    I A Q = Q T
    =>
    A Q = Q T
    =>
    A Q Q^-1 = Q T Q^-1
    =>
    A I = Q T Q^-1
    =>
    A = Q T Q^-1
     
  4. Apr 23, 2009 #3
    Wow, I feel stupid for not noticing that! Thanks
     
  5. Apr 24, 2009 #4
    A is just a change of basis, I'd recommend reviewing change of basis in lin alg
     
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