Let A and D be real n x n matrices.
a) Suppose A is symmetric and has n distinct eigenvalues. Find a two-term expansion of the eigenvalues of the perturbed matrix A + eD, where D is positive definite. What you're finding is known as a Rayleigh-Schrödinger series for the eigenvalues. Hint: Do the same procedure we performed for the Hamiltonians in nondegenerate time-independent perturbation theory. This case is simpler as the operators here are real finite-dimensional matrices.
b) Suppose A is the identity and D is symmetric. Find a two-term expansion of the eigenvalues for the matrix A + √D. Assume you know the solution to the characteristic polynomial of D.