Now prove that the solution of

$\displaystyle argmin_{H \in \mathbb{R}^{kxn}} \{Tr(H M H^T | HH^T = \mathbb{I}_k)\}$

are the first $\displaystyle k$ eigenvectors $\displaystyle u_i$ of a symmetric matrix $\displaystyle M$: $\displaystyle H^T = (u_1 , ... , u_k)$

Hint:

Use the fact that $\displaystyle M$ is a symmetric matrix and therefore has the representation $\displaystyle M = U A U^T$, where $\displaystyle U$ is an orthogonal matrix $\displaystyle UU^T = \mathbb{I}$ where the columns of $\displaystyle U$ contain the eigenvectors of $\displaystyle M$ and $\displaystyle A$ is a diaogonal matrix that contains the eigenvalues on the diagonal. Then use the lemma.