mathematics in reflection operator about complex_rnn

[martinarjovsky]

A unitary matrix is defined as a matrix that has UU* = U*U = I with * the conjugate transpose. If you take the transpose instead of the conjugate transpose in your vector the transformation is not unitary.

Reflection by a complex vector uses therefore the dagger operator (or the conjugate transpose if you want to call it like that). Are you positive the code isn't doing that? I'll check the math in detail later.

[corcra]

Thanks for the quick response. I see, of course the operator should be unitary! Does one still have the geometric interpretation of a rotation in this case, though?

Regarding the maths: to get the expression for the unitary version, it should (I think!) be sufficient to just change the sign of the imaginary part of the reflection (v_im) whenever it appeared in a dot product. Assuming the rest of the algebra is correct, this should result in: (compare with equation 2)

... which still doesn't reproduce the expressions for d-c and a+b! What am I missing here?

[corcra]

I figured it out btw, your code is actually implementing the operator

... which is also unitary, and just a reflection in the conjugate of v, so it's all good.