Suppose A is a symmetric N × N matrix with eigenvectors vi, i = 1, 2, 3… N with
corresponding eigenvalues λi, i = 1, 2,3 …N.
Pick any two distinct eigenvalues (assuming such a pair exists). Let's call them λ1 and λ 2 and their corresponding eigenvectors v1 and v2.
(a) Write down the matrix equations that show that v1 and v2 are eigenvectors of A.
(b) Compute the transpose of the equation satisfied by v2.
(c) Multiply, from the right, the result of part (b) by v1.
(d) Use the assumptions that A is symmetric and λ1≠ λ 2 to deduce a value for v2^t v1.
(e) What important property can you deduce from your result in part (d)?