Let A be an n×n symmetric matrix, let M and m denote the maximum and minimum values of the quadratic form xᵀAx, where xᵀx=1, 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=xᵀAx. Verify that t=(1−α)m+αM for some number α between 0 and 1. Then let x=√1−α​unₙ+√α​u₁​ and show that xᵀx=1 and xᵀAx=t.