Chapter 6.4, Problem 26E

 Let A be a Hermitian matrix with eigenvalues ${\lambda }_1,...,{\lambda }_n$ and orthonormal eigenvectors $u_1,...,u_n.$  Show that
$A={\lambda }_1{u}_1{u}_1^H+{\lambda }_2{u}_2{u}_2^H+\cdots +{\lambda }_n{u}_n{u}_n^H$