This paper provides new results on the application of semidefinite optimization to satisfiability by studying the connection between semidefinite optimization and minimal unsatisfiable subformulas. We use a semidefinite least squares problem to assign weights to the clauses of a propositional formula. We then show that these weights are a measure of the importance of each clause in rendering the formula unsatisfiable. In particular, we show that this approach identifies two important cases: first, if an unsatisfiable formula is a minimal unsatisfiable subformula, then all of its clauses have the same weight; second, if a clause does not belong to any minimal unsatisfiable subformula, then its weight is zero. The results in this paper establish the basis for a semidefinite optimization-based algorithm to identify minimal unsatisfiable subformulas within a given formula. An additional contribution of this paper is a demonstration of how the infeasibility of a semidefinite optimization problem can be tested using a semidefinite least squares problem by extending an earlier result for linear optimization. The connection between the semidefinite least squares problem and Farkas' Lemma for semidefinite optimization is also discussed.