Positive Definiteness Constraint
A matrix $A$ is positive semidefinite if and only if there exists a matrix $V$ such that $$A = V^\top V.$$
So, you can use the entries of $V$ as your unknowns, and express each entry of $A$ as a quadratic function of the unknowns.  Whenever you want to use $A$, instead rewrite that equation in terms of the entries of $V$.