The Diag function sends a vector x in R^n to the diagonal nxn matrix with the components of x along the diagonal. The Diag function is a linear map from R^n to S^n_+, the set of symmetric positive semidefinite matrices.
Since it is a linear map between two inner product spaces, a natural question would be to ask: what is the adjoint operator of the Diag function? I guess that it is the diag function, which takes a matrix and outputs the vector of diagonal entries. Is that true?
The adjoint of a linear map $\mathcal A$ between two inner product spaces $X$ and $Y$ is another linear map denoted $\mathcal A^*$, satisfying $$\langle \mathcal A(x),y\rangle = \langle x, \mathcal A^*(y) \rangle ,$$ for any $x \in X$ and $y \in Y$.