Let A be an n n symmetric matrix. The optimization problem to maximize xTAx subject to x=1 has a solution v1 . v1 is an eigenvector of A.

Then consider the optimization problem:

maximize xTAx subject to x=1 and <x,v1>=0. This problem has 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>=0 and <x,v2>=0 . This problem has a solution v3.

Show that v3 is an eigenvector of A that is orthogonal to v1 and v2 .

Thank you!