Some identities resulting from the Cayley-Hamilton theorem are derived. Some applications include: (a) for k = 1,2,…, n − 1 a condition is found for a pair ( A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of ( A,B) is isomorphic to a subfield of a rational function field with n( n+3)/2 generators consisting of elements symmetric with respect to the permutaion group P n ; (c) it is shown that any rational invariant of ( g+2) symmetric operators A, B, C 1, C 2,…, C g can be expressed as a rational function of invariants of one or two operators that are taken for pairs ( A, B), ( A, C 2),…, ( A, C g , ( A, B+ C 1), ( A, B+ C 2),…,( A, B+ C g ).