In this problem we explore a little further the analogy

Chapter 11, Problem 27

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In this problem we explore a little further the analogy between SturmLiouville boundaryvalue problems and Hermitian matrices. Let A be an n n Hermitian matrix witheigenvalues 1, ... , n and corresponding orthogonal eigenvectors (1), ... , (n).Consider the nonhomogeneous system of equationsAx x = b, (i)where is a given real number and b is a given vector. We will point out a way of solvingEq. (i) that is analogous to the method presented in the text for solving Eqs. (1) and (2).(a) Show that b = ni=1bi(i), where bi = (b, (i)).(b) Assume that x = ni=1ai(i) and show that for Eq. (i) to be satisfied, it is necessary thatai = bi/(i ). Thusx = ni=1(b, (i))i (i), (ii)provided that is not one of the eigenvalues of A, = i for i = 1, ... , n. Compare thisresult with Eq. (13).Greens7 Functions. Consider the nonhomogeneous system of algebraic equationsAx x = b, (i)where A is an n n Hermitian matrix, is a given real number, and b is a given vector. Insteadof using an eigenvector expansion as in 27, we can solve Eq. (i) by computing theinverse matrix (A I)1, which exists if is not an eigenvalue of A. Thenx = (A I)1b. (ii)