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

and

and denote by

the rows of

. Then

(Tr is the trace of a matrix)

Quote:

Now prove that the solution of

are the first

eigenvectors

of a symmetric matrix

:

Hint:

Use the fact that

is a symmetric matrix and therefore has the representation

, where

is an orthogonal matrix

where the columns of

contain the eigenvectors of

and

is a diaogonal matrix that contains the eigenvalues on the diagonal. Then use the lemma.

Thank you very much!