How do I apply the definition of adjoint operator in this problem? U and V are two arbitrary operators, not necessarily Hermitian. Show that (UV )† = V †U†.
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You should be more sincere next time while asking questions,it is not at all for very easy problems that you can answer by little thinking. – Kishalay Sarkar Oct 10 '19 at 10:59
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I'll keep that in mind. Thanks. – RK Ali Oct 10 '19 at 11:11
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We have $$\langle (UV)^Tx, \, y\rangle = \langle x,\, UVy\rangle =\langle U^Tx, \, Vy\rangle =\langle V^T(U^Tx), \, y\rangle$$ Substituting the standard basis vectors $e_i$ and $e_j$ for $x$ and $y$ yields that the $i,j$ entries of $(UV)^T$ and $V^TU^T$ are the same.
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thank you so much. i have confirmed my answer. I used (u,v) instead of (x,y). – RK Ali Oct 10 '19 at 10:39