Will a vector rotate back to itself under unitary rotation?Let there be an arbitrary complex vector v → ∈ C D that has unit length defined as ∑ i v i v i ∗ = 1. Let there be a Hermitian D × D matrix H. Given arbitrary v → and H, will there always exist a non-zero t ∈ R such that e i H t v → = v → ?

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Will a vector rotate back to itself under unitary rotation?Let there be an arbitrary complex vector             v      →        ∈            C        D   that has unit length defined as       ∑    i        v    i        v    i    ∗    =  1. Let there be a Hermitian   D  ×  D matrix H. Given arbitrary             v      →        and H, will there always exist a non-zero   t  ∈      R   such that       e          i      H      t                  v      →        =            v      →      ?

klunnalegj6 · Accepted Answer

No. Consider   H  =      (                            1                          0                                      0                                      3                                )  . Then      e          i      H      t        =      (                                        e                          i              t                                                0                                      0                                      e                          i                              3                            t                                            )  And it is now apparent that the vectors   v  ∈            C        2   such that there is some   t  ≠  0 satisfying       e          i      H      t        v  =  v are the ones such that either       v    1    =  0 or       v    2    =  0Added: In the same spirit, it isn&#039;t hard to show that the hermitian matrices H such that for all unitary vectors v there is some   t  ∈      R    ∖  {  0  } such that       e          i      t      H        v  =  v are exactly the ones such that there is some real number   α such that       λ    α    ∈      Q   for all   λ eigenvalues of H.More precisely, let   H  =      H    ∗    ∈            C              n      ×      n       and   v  ∈            C        n  . Let&#039;s call       V    λ    =  ker  ⁡  (  H  −  λ  I  ) and let&#039;s impose the following equivalence relation on   spec  ⁡  H:  λ  R  μ    ⟺    λ  =  μ  =  0  ∨      (    λ    ≠    0    ∧    μ    ≠    0    ∧          μ      λ        ∈          Q        )  Let&#039;s call       W          λ        =      ⨁          μ      R      λ            V    μ  . Then there is some   t  ∈      R    ∖  {  0  } such that       e          i      t      H        v  =  v if and only if   v  ∈      V    0    +      W    λ   for some   λ  ∈  spec  ⁡  H

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