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Is it valid to think of complex unitary matrix as some rotation in complex vector space analogous to real unitary matrix as some rotation in real vector space? The reason I ask this is because I tired to find a 2x2 complex unitary matrix which can perform certain rotation to vectors in C2 space, specifically, I want to rotation [a1+i0, a2+i0]^T to [0+i0, a3+ib]^T (rotate a vector in the R2 subspace of C2 to C1 subspace of C2). I thought i was possible because if you think of imaginary axis just as additional dimension, this rotation would have no difference from rotating a vector in R4 which initially in a R2 subspace to another R2 subspace. But the numerical method I used suggested me such a 2x2 complex unitary matrix does not exist.

Sam
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  • No. Orthogonal transformations in $\mathbb{R}^4$ (i.e. rotations and reflections) satisfy $A^TA=I$, which does not become $A^*A=I$ when you rewrite them as transformations in $\mathbb{C}^2$. There are many more $\mathbb{R}^4$ rotations. – Conifold Apr 16 '20 at 00:34
  • By 'no' do you mean complex unitary matrix cannot be interpreted as rotation in complex space? – Sam Apr 16 '20 at 01:19
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    It can, see What's the interpretation of a unitary matrix? But there are rotations not representable by such matrices, including the one you want. – Conifold Apr 16 '20 at 01:35
  • sorry I forgot to mention that [0+i0, a3+ib]^T has the same norm defined as inner product of complex vectors (conjugate transpose of one vector dot with another) as [a1+i0, a2+i0]^T. So if assuming they have the same norm, why I cannot map the first one to the second one by some rotation? – Sam Apr 16 '20 at 10:17

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