Show that the eigenvalues of the overlap matrix are positive

[jarra]

Homework Statement

The task is to show that the eigenvalues of overlap matrix $$\tilde S$$ are positive.

Homework Equations

The overlap matrix is defined as $$(\tilde S)_{nm} = \langle \xi_n \vert \xi_m \rangle$$, with $$\xi_k$$ being the base vectors of the wavefunction. http://en.wikipedia.org/wiki/Overlap_matrix

The Attempt at a Solution

I've tried to show that the eigenvalues are positive by showing that $$\tilde S$$ is positive definite. Both with the condition $$\vec x^{\ast} \tilde S \vec x > 0$$ and the condition that all the 'sub-determinants' are larger than zero. But I don't get it right, please help.

 

[Marco_84]

Homework Statement

The task is to show that the eigenvalues of overlap matrix $$\tilde S$$ are positive.

Homework Equations

The overlap matrix is defined as $$(\tilde S)_{nm} = \langle \xi_n \vert \xi_m \rangle$$, with $$\xi_k$$ being the base vectors of the wavefunction. http://en.wikipedia.org/wiki/Overlap_matrix

The Attempt at a Solution

I've tried to show that the eigenvalues are positive by showing that $$\tilde S$$ is positive definite. Both with the condition $$\vec x^{\ast} \tilde S \vec x > 0$$ and the condition that all the 'sub-determinants' are larger than zero. But I don't get it right, please help.

In a naively way i would say: It is just an integral of a probability distribution---> it is positive by definition!

More thecnically: the diagonal members are for sure positive and what about the off diagonal?

Hint: use the fact that $$S_{jk}=\overline{S}_{kj}$$