Minimizing the trace of a special matrix

Hi everybody,

I don't know how to solve the following problem. I have rewritten the things according to the hint, but couldn't make much use of it.

Quote:

Lemma:

Let $\displaystyle H \in \mathbb{R}^{k x n}$ and $\displaystyle L \in \mathbb{R}^{n x n}$ and denote by $\displaystyle h_i , i = 1,...,k$ the rows of $\displaystyle H$. Then

$\displaystyle Tr(H L H^T) = \sum_{i=1}^k <h_i , L h_i>$

(Tr is the trace of a matrix)

Quote:

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.

Thank you very much!