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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?