I How can I test for positive semi-definiteness in matrices?

Trollfaz
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On a side note I'm posting on PF more frequently as I have exams coming and I need some help to understand some concepts. After my exams I will probably go inactive for a while.
So I'll get to the point. Suppose we have a matrix A and I wish to check if it is positive semi definite. So one easy way is to see if all it's eigenvalues are ##\ge 0##.
Another way is to test using the definition of PSD
$$v^T Av\ge 0\ v \in R^n$$
But sometimes things get really messy when I try to test a matrix with arbitrary parameters say I'm testing ##\triangledown ^2 f(x)## for PSD to check if f(x) is convex. Is there any other ways to prove for PSD in a matrix
 
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Do you have a specific example of a problem you're stuck trying to solve? I don't think there's any general principle beyond what you listed but an example might spark some specific insight or just help demonstrate how to use the definition to check.
 

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