Show that the eigenvalues of the overlap matrix are positive

In summary, the task is to show that the eigenvalues of the overlap matrix \tilde S are positive. 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. Various attempts have been made to prove this, using conditions such as \vec x^{\ast} \tilde S \vec x > 0 and ensuring that all 'sub-determinants' are larger than zero. However, it can be easily shown that the overlap matrix is positive definite, as it is just an integral of a probability distribution and, therefore, is positive by definition. Additionally, using the fact that S
  • #1
jarra
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Homework Statement


The task is to show that the eigenvalues of overlap matrix [tex]\tilde S[/tex] are positive.


Homework Equations


The overlap matrix is defined as [tex] (\tilde S)_{nm} = \langle \xi_n \vert \xi_m \rangle [/tex], with [tex]\xi_k[/tex] 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 [tex]\tilde S[/tex] is positive definite. Both with the condition [tex]\vec x^{\ast} \tilde S \vec x > 0[/tex] and the condition that all the 'sub-determinants' are larger than zero. But I don't get it right, please help.
 
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  • #2
jarra said:

Homework Statement


The task is to show that the eigenvalues of overlap matrix [tex]\tilde S[/tex] are positive.


Homework Equations


The overlap matrix is defined as [tex] (\tilde S)_{nm} = \langle \xi_n \vert \xi_m \rangle [/tex], with [tex]\xi_k[/tex] 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 [tex]\tilde S[/tex] is positive definite. Both with the condition [tex]\vec x^{\ast} \tilde S \vec x > 0[/tex] 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 probabilty 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 [tex]S_{jk}=\overline{S}_{kj}[/tex]
 

Related to Show that the eigenvalues of the overlap matrix are positive

What is an overlap matrix?

An overlap matrix is a square matrix used in quantum mechanics to represent the overlap between two wavefunctions. It is also known as the S matrix or the matrix of overlap integrals.

Why is it important to show that the eigenvalues of the overlap matrix are positive?

In quantum mechanics, the eigenvalues of the overlap matrix represent the probability of finding a particle in a particular state. If the eigenvalues are negative, it would mean that the probability is negative, which is physically impossible. Therefore, it is important to show that the eigenvalues are positive to ensure the validity of the results.

How can I show that the eigenvalues of the overlap matrix are positive?

The eigenvalues of the overlap matrix can be shown to be positive by using mathematical techniques such as diagonalization or the Gershgorin circle theorem. Additionally, certain properties of the overlap matrix, such as symmetry and positive definiteness, can also be used to prove that the eigenvalues are positive.

What does it mean if the eigenvalues of the overlap matrix are not all positive?

If some of the eigenvalues of the overlap matrix are not positive, it could indicate an error in the calculations or an issue with the wavefunctions being used. It is important to identify and address these discrepancies in order to obtain accurate results in quantum mechanical calculations.

Are there any other matrices with similar properties to the overlap matrix?

Yes, there are other matrices in quantum mechanics that share similar properties to the overlap matrix, such as the Hamiltonian matrix and the density matrix. These matrices also have eigenvalues that represent physical quantities and must satisfy certain conditions, such as positivity, to ensure the validity of the results.

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