We consider the following relation on the set of real symmetric matrices of format $(2,2)$: we say that $S \leqslant S ^ { \prime }$ if and only if the symmetric matrix $S ^ { \prime } - S$ satisfies $S ^ { \prime } - S \geqslant 0$ (i.e., all eigenvalues of $S' - S$ are $\geqslant 0$).
Prove that the relation $\leqslant$ above is indeed an order relation on real symmetric matrices of format $(2,2)$.