Thank you for taking the time to read my question. I've had a look at other posts regarding differentiation of Hadamard Products of matrices but none of the examples are the same as the one I am curious about. I am looking to do the following:
$ \frac{dL}{d\beta} $ where $L = y'(X\beta \: ⊙ \: X\beta ) $ where y is a N by 1, X is N by K and $\beta$ is K by 1. Denote $\underline{x_i'}$ the i-th row of the matrix X.
Here are my thoughts : $y'(X\beta \: ⊙ \: X\beta ) = y_1(\underline{x_1'}\beta)^{2}+...+y_n(\underline{x_n'}\beta)^{2}$ so its derivative for any entry $\beta_k$ in $\beta$ will be equal to $2y_1 x_{1,k}\underline{x_1'}\beta \: + \: ... +2 y_n x_{n,k}\underline{x_n'}\beta = 2<y⊙\underline{x_k}, X\beta >$ which suggests that, in matrix form, it should be something like $2y'X ⊙ X\beta$ ? This is only my intuition.
Many thanks for any help and time you may provide.