Let A be an n x n symmetric matrix, let M and m denote the maximum and

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^Tx, where x^Tx = 1, and denote corresponding unit eigenvectors by u₁and u_n. 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 α between 0 and 1. Then let x = √1 - αu_n+ √αu₁, and show that x^Tx = 1 and x^TAx = t.