When is a matrix positive semi-definite?

[JK423]

Hello people,

Im working on a project and this problem came up:

I have a symmetric matrix whose elements are complex variables, and i know that this matrix is positive semi-definite.
I have to derive a criterion for the matrix's elements, so that if it's satisfied by them then the matrix will be positive semi-definite.

Any idea on how to do that?
For example, a positive semi-definite matrix has to satisfy some relation that i can use?
Maybe its eigenvalues must be non-negative?

I'd really need your help, thanks a lot!

John

 

[HallsofIvy]

Yes, a matrix is "positive semi-definite" if and only if all of its eigenvalues are non-negative. You might want to look at this: http://en.wikipedia.org/wiki/Positive-definite_matrix

 

[micromass]

Yes, a matrix is "positive semi-definite" if and only if all of its eigenvalues are non-negative. You might want to look at this: http://en.wikipedia.org/wiki/Positive-definite_matrix

You'll want the matrix to be Hermitian as well (or normal).

 

[AlephZero]

You'll want the matrix to be Hermitian as well (or normal).

Complex symmetric (not hermitian!) matrices do occur in modelling processes which don't conserve energy, but then the concept of "positive semidefinite" isn't very meaningful. Ths signs of the real parts of the eignenvalues is usually more interesting physically - i.e. does the energy of the system increase or decrease.

 

[blue_raver22]

Hello John,

A matrix is positive semi-definite if all of its eigenvalues are non-negative. This means that if you can find a way to ensure that the eigenvalues of your matrix are non-negative, then you can conclude that the matrix is positive semi-definite. One way to do this is by using the Sylvester's criterion, which states that a Hermitian matrix (a matrix whose elements are complex variables and is equal to its own conjugate transpose) is positive semi-definite if and only if all of its principal minors (submatrices formed by taking the first k rows and columns) have non-negative determinants. In other words, if all the submatrices formed by taking the first 1, 2, 3, ..., n rows and columns have non-negative determinants, then the matrix is positive semi-definite. This can be a useful criterion to check for positive semi-definiteness of a matrix with complex elements. I hope this helps! Good luck with your project.

Best,