I have been reading about semidefinite programs and I feel completely lost as to what is going on. The lagrangian multipliers for linear, geometric and second order cone programs can be expressed as a scalar, multiplied with each constraint, or a dot product between a lagrangian vector and all the constraints $Ax - b$. And I understand this perfectly because a positive lagrangian for a $\le$ inequality constraint results in a higher value if the constraints are violated (since we are trying to minimize).
Question: In semidefinite programs, why are the lagrangian multipliers a matrix ? And is it also symmetric / psd ? I understand that for a SDP, the inequality constraint needs to be positive semidefinite but I don't understand how multiplying the constraints with a lagrangian-matrix encodes this constraint like in a LP, GP, SOCP. Im pretty sure im lacking some basic knowledge because I have 0 clue as to what is going on here.
Below is an example of a SDP whose dual is shown here https://neos-guide.org/content/semidefinite-programming. I have been trying to derive that but I don't know where to start.
Here is another example that I can't follow.
