For a SDP problem with LMI, we can write it as: $$minimize \quad c^Tx$$ $$ s.t. \quad x_1F_1+x_2F_2+...+x_nF_n+G \preceq 0$$ $$Ax=b$$ where $G,F_1,F_2,...,F_n \in S^k,A \in R^{p \times n}$.
Now I want to transform above form to SDP of standard form: $$minimize \quad tr(CX)$$ $$s.t. \quad tr(A_iX)=b_i,i=1,2,...,p$$ $$X \succeq 0$$ Where $C,A_1,...,A_p \in S^n$.
Suppose $A=\begin{bmatrix} a^T_1 \\ a^T_2 \\ \vdots \\a^T_p\end{bmatrix}$, I tried to diagonalize $c,x,a_i$ such that: $$C=diag(c),X=diag(x),A_i=diag(a_i)$$
But I cannot figure out how satisfying constraint $X \succeq 0$. So what is the trick?