One way, probably not the best, would be to check the eigenvalues of the matrix difference.
matrixGreaterEqual[m1_, m2_] /;
MatrixQ[m1, Element[#, Reals] &] &&
MatrixQ[m2, Element[#, Reals] &] &&
Dimensions[m1] == Dimensions[m2] && Apply[Equal, Dimensions[m1]] :=
Module[{diffmat = m1 - m2, evals},
evals = Eigenvalues[diffmat];
VectorQ[evals, Element[#, Reals] &] && Min[evals] >= 0]
Example:
m1 = {{5, -1}, {-1, 3}};
m2 = {{2, -1}, {-1, 2}};
In[11]:= matrixGreaterEqual[N@m1, m2]
(* Out[11]= True *)
There is probably a better way to go about this using CholeskyDecomposition.
