Morse index and degeneracy

Question

The function is as follows: $$f(x,y,z) = e^x(xy-y^2-z^2)$$ I have found the critical points to be $(0,0,0)$ and $(-2,1,0)$. The question asks to determine the morse index of the points and the degeneracy. How am I supposed to determine the morse index? Also, what exactly is a degenerate critical point? How do I determine whether it is a local maximum or local minimum?

Answer 1

At a critical point of a function $f$ (that is, points where $\nabla f = 0$), its Hessian is a well-defined bilinear form and is independent of coordinate changes. By Sylvester's law of inertia, to each bilinear form on a vector space we can associate three numbers: the dimensions of the maximum subspaces on which the bilinear form is positive definite, zero, or negative definite.

The Morse index of a critical point is the negative index of inertia. In other words, it is the dimension of the maximum subspace on which the Hessian is negative definite.

The degeneracy refers to the maximum dimension of the zero subspace. If this dimension is nonzero, the critical point is said to be degenerate; and it is nondegenerate otherwise.

Therefore, to answer all your questions, you need to compute the Hessian matrix of $f$ at the two critical points. For each of the two matrices, you have to find its indices of inertia (see the Wikipedia page above). Based on that information you can read off the Morse index and degeneracy.

For a nondegenerate critical point, it is a local maximum if and only if the positive index of inertia is zero. Alternatively, for a nondegenerate critical point, it is a local minimum if and only if the Morse index is zero. If the critical point is degenerate, then based on Hessian alone you cannot determine whether the critical point is a local maximum or minimum (but you can tell when it is not: if both the positive and negative indices are nonzero, then you know the critical point must be a saddle point).