Orthogonality Constraints

A continuation of Constrained Optimization

Orthogonality Constrained Optimization

If $Q=x^{T}Ax$, $A$ is a real $n\times n$ symmetric matrix, with eigenvalues ${\lambda }_1\ge {\lambda }_2\cdots \ge {\lambda }_n$ and associated normalized eigenvectors $u_1,u_2,\dots ,u_n$ Also, $||x||=1$ AND $x\cdot u_1=0$ Then max value of $Q(x)={\lambda }_2$ attained at $u_2$ Then min value of $Q(x)={\lambda }_n$ attained at $u_n$