Question: Let A be an n X n symmetric matrix, let M and m denote the maximum and minimum values of the quadratic form x TAx, and denote corresponding unit eigenvectors by u1 and un. The following calculations show that given any number t between M and m, there is a unit vector x such that t = x TAx. Verify that t = (1 - α)m + αM for some number ? between 0 and 1. Then let x = √1 - αun + √αu1, and show that x Tx = 1 and x TAx = t.