Let be a linear operator that is self-adjoint with respect to some inner proudct . Then the -th largest eigenvalue of is

where the infimum is taken over all -dimensional subspaces and the supremum is taken over all with .

Let be a linear operator that is self-adjoint with respect to some inner proudct . Then the greatest eigenvalue satisfies

For a symmetric matrix and vector , the eigenvalues of A interlace the eigenvalues of , i.e.

Let be a symmetric matrix, and let . If has at most one positive eigenvalue, then has at most one positive eigenvalue.

Let and . Define for each . Then