Write out the details in the proof of Theorem 8.30.
Theorem 8.30
Let A be a symmetric matrix with eigenvalues λ1 ≥ λ2 ≥ ... ≥ λn and corresponding orthogonal eigenvectors v1,..., vn. Then the maximal value of the quadratic form q(x) = xT A x over all unit vectors that are orthogonal to the first j - 1 eigenvectors is its jth eigenvalue:
λj = max {xT Ax | ||x|| = 1, x ∙ v1 = ... = x ∙ vj-1 = 0 }. (8.39)
Thus, at least in principle, one can compute the eigenvalues and eigenvectors of a symmetric matrix by the following recursive procedure. First, find the largest eigenvalue λ1 by the basic maximization principle (8.37) and its associated eigenvector v1 by solving the eigenvector system (8.13). The next largest eigenvalue λ2 is then characterized by the constrained maximization principle (8.39), and so on. Although of theoretical interest, this algorithm is of somewhat limited value in practical numerical computations.