(This question is a way to prove the spectral theorem)

Let A be an n n symmetric matrix. The optimization problem is maximize xTAx subject to ||x||=1. It admits a solution v1 . Using Lagrange multipliers, we know v1 is an eigenvector of A. Now consider the optimization problem: maximize xTAx subject to ||x||=1 and <x,v1>=0. This problem admits a solution v2 .

a) Show that v2 is an eigenvector of A that is orthogonal to v1.

b) Consider the optimization problem: maximize xTAx subject to ||x||=1 and <x,v1>=0and <x,v2>=0 . This problem admits a solution v3. Show that v3 is an eigenvector of A that is orthogonal to v1 and v2.

Thank you!