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Math Help - Adjoint of a transformation

  1. #1
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    Adjoint of a transformation

    Let V be the set of all complex square matrices. For D \in V define T_D : V \rightarrow V by T_D(A) = DA.

    Using the inner product <X,Y>=tr(XY^*), find the adjoint of T_D.

    I feel I'm missing something because I can't arrive at an answer.
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  2. #2
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    The adjoint T_D^* of T_D satisfies the condition \langle T_D^*(A),B\rangle = \langle A,T_D(B)\rangle = \langle A,DB\rangle for all B, where the angled brackets denote the inner product. If you write this in terms of the trace then it becomes \text{tr}(T_D^*(A)B^*) = \text{tr}(A(DB)^*) = \text{tr}(AB^*D^*). But the trace has the property that \text{tr}(XY) = \text{tr}(YX). This means that \text{tr}(T_D^*(A)B^*) = \text{tr}(D^*AB^*). From that you should be able to see what T_D^*(A) is.
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