# Schur decomposition ambiguity

junglebeast
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?

Werg22
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

junglebeast
Wow, I feel stupid for not noticing that! Thanks

samspotting
A is just a change of basis, I'd recommend reviewing change of basis in lin alg