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.

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

Tr(H L H^T) = \sum_{i=1}^k <h_i , L h_i>
(Tr is the trace of a matrix)
Now prove that the solution of
argmin_{H \in \mathbb{R}^{kxn}} \{Tr(H M H^T | HH^T = \mathbb{I}_k)\}
are the first k eigenvectors u_i of a symmetric matrix M: H^T = (u_1 , ... , u_k)
Hint:
Use the fact that M is a symmetric matrix and therefore has the representation M = U A U^T, where U is an orthogonal matrix UU^T = \mathbb{I} where the columns of U contain the eigenvectors of M and A is a diaogonal matrix that contains the eigenvalues on the diagonal. Then use the lemma.
Thank you very much!