From Wikipedia
Denote by $\mathbb{S}^n$ the space of all $n \times n$ real symmetric matrices. The space is equipped with the inner product (where ${\rm tr}$ denotes the trace) $$\langle A,B\rangle_{\mathbb{S}^n} = {\rm tr}(A^T B) = \sum_{i=1,j=1}^n A_{ij}B_{ij}.$$ We can rewrite the mathematical program given in the previous section equivalently as $$ \begin{array}{rl} {\displaystyle\min_{X \in \mathbb{S}^n}} & \langle C, X \rangle_{\mathbb{S}^n} \\ \text{subject to} & \langle A_k, X \rangle_{\mathbb{S}^n} \leq b_k, \quad k = 1,\ldots,m \\ & X \succeq 0 \end{array} $$
From Boyd's paper, a semidefinite programming problem is
$$\min_{x \in\mathbb R^m} c^T x$$ subject to $$F_0 + \sum_{i=1}^m x_i F_i ⪰ 0$$ where $c \in \mathbb R^m$ and $m + 1$ symmetric matrices $F_0, ..., F_m \in \mathbb R^{n\times n}$.
I was wondering if the two formulations are equivalent? I am not able to see how they are related. Thanks and regards!