Why do we say that the eigenspaces of unitary matrices are orthogonal?

Question

The matrix http://www.wolframalpha.com/input/?i=(sqrt(5)%2F5)((1,2),(2i,-i)) is unitary.

The eigenvalues are listed, as are a pair of eigenvectors associated to these eigenvalues.

http://www.wolframalpha.com/input/?i=(-1%2Bi,2)+dot+product+(1-i,1)

They are not orthogonal, as the dot product is not 0.

How is this not a counter-example?

Thank you.

Answer 1

Wolfram alpha is not computing the complex dot product correctly. The standard dot product is $v^T\bar{w}$ where $\bar{w}$ is component-wise complex conjugation. However, Wolfram alpha computes $v^Tw$ without complex conjugation. If do the conjugation ourselves, we do get the the right result.