# Minimizing the trace of a special matrix

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• Feb 5th 2009, 02:11 AM
Kyle_Katarn
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 $H \in \mathbb{R}^{k x n}$ 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 $
(Tr is the trace of a matrix)
Quote:

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!