Since somebody here told me that it is in general insufficient to show that a matrix is positive definite when all eigenvalues are positive. I am interested in finding good ways to prove this. In general, it might be hard to verify the definition. So, I am looking for different ways to do this. Probably, I should point out that if you have an answer that works only in $\mathbb{R}$ or $\mathbb{C}$ vector spaces, I would highly appreciate such remarks too.
Thanks in advance!
Because the question came up what I mean by positive definite: No, I do not assume a positive definite matrix to be necessarily hermitian and Yes I want to have a fast way to see that a given matrix is positive definite.
Best Answer
For a Hermitian matrix (over $\mathbb R$ or $\mathbb C$) it is indeed necessary and sufficient that all eigenvalues are positive in order for the matrix to be positive definite, see e.g. http://en.wikipedia.org/wiki/Positive-definite_matrix#Characterizations . I haven't heard of positive definite used to describe any operator that is not Hermitian. It is easy to check that a matrix is Hermitian. If you don't want to compute all eigenvalues to verify they're positive, you can use Sylvester's criterion instead. It says that a Hermitian matrix is positive definite is all upper-left blocks (including the full matrix) have positive determinants. The link also gives other characterizations, but they may be harder to check in practice.