The usual way to define a semi-definite program (SDP), e.g., as given in Boyd and Vandenberghe's convex optimization book, is: $$ \begin{array}{cl} \min & c^\top x \\ \mathrm{s.t.} & 0 \succeq x_1 F_1+ \ldots+x_nF_n + G \\ & Ax = b \end{array} $$ where $ 0 \succeq M$ denotes $0 - M$ is a p.s.d. matrix. In the above problem $x \in \mathbb{R}^{n}$ is an optimization variable, while $F_1,\ldots,F_n, G$ are symmetric (known) matrices. Similarly $A \in \mathbb{R}^{m \times n},b \in \mathbb{R}^n$ are (known) linear equality system's matrices.
My question is, without being bothered by the standard notation as above, can we say any problem which has all linear constraints and an additional p.s.d.-ness constraint is SDP? For example, $$ \begin{array}{cl} \min & c^\top x \\ \mathrm{s.t.} & \sum_{(j,k) \in S_i} Y_{j,k} = x_i, \quad i=1,\ldots,n\\ & Y \succeq 0 \end{array} $$ Here, $S_i$ is a set of indexes, and for each $i$ we are summing the elements $(j,k) \in S_i$ of matrix $Y$ and equate it to $x_i$.
The reason I am confused is that writing $Y$ in the latter problem in form $x_1 F_1+ \ldots+x_nF_n + G$ seems not straightforward, i.e., another linear system on $Y$ matrix has to be solved, etc..
