Suppose I have a 50-dimensional field. I compute the Hessian matrix at a stationary point and find 40 negative eigenvalues + 10 positive ones. Can I conclude that the point is "mostly" a maximum with 40 downhill directions but only 10 uphill ones, and further that the most-negative eigenvalue corresponds to the direction of fastest descent? If the 50-dimensional field takes a form like $F = f(x) + f(y) + f(z) + f(a) + ...$ where all the terms are independent of each other, then both statements are trivially true, but I'm wondering if it's true in general.

Intuitively the answer seems like yes, but I've not been able to find this stated explicitly. There're a lot of sources that discuss the interpretation (e.g. page 13-14 of this), but they're usually in 2D where saying there's a saddle is exactly the same as saying there's one uphill and one downhill direction. Among those that mention 3D (e.g. this), they generally just say it's a saddle and leave it at that. None of the sources discuss the magnitudes of the eigenvalues either, only the sign.


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