Let $F: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function. What do the eigenvalues of $H(f)$ tell us? In particular, if $H(F)$ has both positive and negative eigenvalues at a critical point $(x_1,\ldots,x_n)$, does $F$ have a "saddle point"?
Meaning of hessian for vector valued functions
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calculus
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0The "hessian" isn't a matrix but a rank 3 tensor... – 2017-02-22
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0Quite right, my mistake. In any case, what is the meaning of this tensor? How can we use it to discuss the geometry of the graph? – 2017-02-22
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0Rank 3 tensors don't have the concept of eigenvalues. As for geometry of $F$, just analyze its components separately. – 2017-02-22