Adjoint of a transformation

**0-)** · November 21st 2008, 05:33 AM

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.

**Opalg** · November 21st 2008, 11:54 AM

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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