So I understand that the change of basis of the matrix is $X' = U^{\dagger}XU$. But how about if I just want to change the basis of a vector $v$.
Is $v' = U^{\dagger}v$ the right one? Why does the $U$ disappear here?
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You are using a lot of symbols without letting us know what any of them mean. – Gerry Myerson Sep 23 '16 at 07:18
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related: http://math.stackexchange.com/questions/340978/change-of-basis-matrix – Ka Wa Yip Sep 23 '16 at 07:33
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Presumably $U$ is unitary and $\dagger$ is the adjoint. – Ben Grossmann Sep 23 '16 at 11:49
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1The key is to understand that $X'v'$ should produce the same result as $Xv$, but that output $Xv$ of $X$ should be expressed with respect to the new basis. – Ben Grossmann Sep 23 '16 at 11:53
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@Omnomnomnom thank you. does that mean $v' = X'^{\dagger}Xv$? – Ka Wa Yip Oct 06 '16 at 22:06
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It means that $Xv = UX'v'$, which is the same as $UX'(U^\dagger v)$ – Ben Grossmann Oct 06 '16 at 23:04
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@Omnomnomnom does this mean that $v' = U^{\dagger}v$ is correct? thanks. – Ka Wa Yip Oct 07 '16 at 00:00
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Yes ${}{}{}{}{}$ – Ben Grossmann Oct 07 '16 at 04:46