Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.


    Mathworld defines the Shur decomposition of matrix A as

    Q^H A Q = T,

    where Q is unitary and T is upper triangular.


    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook