Non degeneracy of critical points

In summary, a critical point of a function f:M\rightarrowR is considered non-degenerate if its Hessian matrix is invertible. This means that all first derivatives are zero and the point is also known as an "ordinary double point". To show that this definition is independent of the choice of local coordinates, one can use the chain rule to perform a coordinate change and check that the Hessian matrix remains invertible.
  • #1
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In local coordinates what does it mean for a critical point of a function f:M[tex]\rightarrow[/tex]R to be non degenerate?
In addition how can you show that the definition is independent of the choice of
local coordinates?

I know that being a critical point is independent of the choice of local coordinates but
I am struggling with the second derivate in local coordinates.
Any help is appreciated?
 
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  • #2
well at a critical point, all 1st derivatives are zero, and then you have as next best thing, the symmetric "Hessian" matrix of 2nd derivatives. non degenerate means that Hessian matrix is invertible. Such a point is also called an "ordinary double point". The most naive way to check that is an invariant notion is to go in there and slog it out with a coordinate change by the chain rule.
 

1. What is the concept of non-degeneracy of critical points?

The non-degeneracy of critical points refers to a property of a critical point in a mathematical function where the Hessian matrix, which represents the second-order derivatives of the function, is non-singular at that point. In simpler terms, it means that the critical point is not a point of inflection and has a distinct and unique curvature, making it a true maximum, minimum, or saddle point.

2. Why is non-degeneracy of critical points important in optimization problems?

In optimization problems, the goal is to find the maximum or minimum value of a function. Non-degeneracy of critical points ensures that the critical points found are actual extrema and not just points of inflection. This is crucial in accurately determining the optimal solution for a given problem.

3. What conditions must be satisfied for a critical point to be non-degenerate?

For a critical point to be non-degenerate, the Hessian matrix at that point must be non-singular, meaning that its determinant is non-zero. This ensures that the second-order derivatives of the function exist and have a unique curvature at that point.

4. Can a critical point be non-degenerate in one direction but degenerate in another?

Yes, it is possible for a critical point to be non-degenerate in one direction but degenerate in another. This occurs when the Hessian matrix is singular in one direction but non-singular in another. In such cases, the critical point is considered non-degenerate overall.

5. How is the non-degeneracy of critical points determined in practice?

The non-degeneracy of critical points can be determined by calculating the Hessian matrix at the critical point and checking if its determinant is non-zero. If the determinant is zero, the point is degenerate and further analysis is required to determine its nature. Alternatively, software programs and algorithms can be used to automatically check for non-degeneracy of critical points in optimization problems.

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