Bounds on the error of an approximate invariant subspacefor non-self-adjoint matrices

Suppose one approximates an invariant subspace of an(Formula presented.) matrix in(Formula presented.) which in not necessarilyself--adjoint. Supposethat one also has an approximation for the corresponding eigenvalues. Weconsider the question of how good the approximations are. Specifically, wedevelop bounds on the angle between the approximating subspace and theinvariant subspace itself.These bounds are functionsof the following three terms: (1) the residual of the approximations; (2)singular--value separation in an associated matrix; and (3) the goodnessof the approximations to the eigenvalues.