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

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SUMMARY

This discussion focuses on methods to test for positive semi-definiteness (PSD) in matrices. The primary techniques mentioned include checking if all eigenvalues of the matrix are greater than or equal to zero and using the definition of PSD, which states that for a vector v in R^n, the condition vTAv ≥ 0 must hold. The conversation highlights the challenges faced when dealing with matrices that contain arbitrary parameters, such as the Hessian matrix ∇²f(x) when assessing the convexity of a function f(x). The participants emphasize the importance of specific examples to clarify the application of these methods.

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  • Understanding of eigenvalues and eigenvectors
  • Familiarity with matrix notation and operations
  • Knowledge of convex functions and their properties
  • Basic concepts of linear algebra, particularly regarding positive semi-definite matrices
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  • Study the properties of eigenvalues in relation to matrix definiteness
  • Explore the application of the definition of positive semi-definiteness in various contexts
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  • Investigate specific examples of matrices with arbitrary parameters and their PSD verification
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Students in mathematics or engineering, researchers in optimization, and anyone involved in linear algebra who needs to understand the conditions 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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