What is the difference between the two definitions of Schur decomposition?

[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