The inner product of u = col(u1, … un) and v = col(v1, ..., vn) is defined in Section 1.14...

The inner product of u = col(u1, … un) and v = col(v1, ..., vn) is defined in Section 1.14 as u1v1 + + unvn and hence equals u′v = v′u. Also |u| is defined as (u′u)1/2; u and v are termed orthogonal if u′v = 0. Now let A be a real symmetric n × n matrix, so that A = A′. Let λ1, λ2 be eigenvalues of A which are unequal and let v1, v2 be corresponding eigenvectors. Show that v1 and v2 are orthogonal. [Hint: First show that x′Ay = y′Ax for all x, y. Then from Av1 = λ1v1, Av2 = λ2v2 deduce that λ1v2′v1 = λ2v1′v2 and hence that v1′v2 = 0.]