Optimization problems involving the eigenvalues of a real symmetric matrix are common in many areas of research. In particular, we consider the sensitivity of all the eigenvalues of a symmetric matrix under small perturbations. In this thesis we present several technical results on perturbations of invariant subspaces. Applying these results we show that, for any symmetric matrix ' A,' the 'm'-th largest eigenvalue of 'A'+' tE'('t') has a 'k'-th order Taylor expansion at 't' = 0+ when the symmetric matrix ' E'('t') depending on the real parameter 't' has a ('k' - 1)-th order Taylor expansion at 't' = 0+ and derive a formula for the coefficients of this expansion. Using this formula we calculate the first four directional derivatives of the 'm'-th largest eigenvalue of a symmetric matrix ' A' in a direction 'E.'