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## Main Question or Discussion Point

Several questions I have been thinking about... let me know if you have thoughts on any of them I added numbers to for coherence and readability.

So, the Hessian matrix can be used to determine the stability of critical points of functions that act on [itex]\mathbb{R}^{n}[/itex], by examining its eigenvalues, but is there any interpretation of what it means to diagonalize the Hessian?

For example, in diagonalizing the Jacobian of a function [itex]f: \mathbb{R}^{n} → \mathbb{R}^{m}[/itex], there are implications that can be drawn about the transformation, and the matrix itself defines how the points transform under [itex]f[/itex]

i.e. [itex]H(f) = \mathrm{diag}(\frac{∂^{2}f}{∂x_{1}^{2}},\cdots,\frac{∂^{2}f}{∂x_{n}^{2}}) ⇔ f(x_{1},\cdots,x_{n}) = \sum g_{i}(x_{i})[/itex], where [itex]g_{i}[/itex] is some arbitrary function.

So, the Hessian matrix can be used to determine the stability of critical points of functions that act on [itex]\mathbb{R}^{n}[/itex], by examining its eigenvalues, but is there any interpretation of what it means to diagonalize the Hessian?

For example, in diagonalizing the Jacobian of a function [itex]f: \mathbb{R}^{n} → \mathbb{R}^{m}[/itex], there are implications that can be drawn about the transformation, and the matrix itself defines how the points transform under [itex]f[/itex]

**(1)**Since the Hessian is second derivatives, maybe it says something about the Jacobian? But I'm not sure about this since there are more second derivatives than first derivatives.**(2)**I'm wondering that since the Hessian is a matrix, can be thought of as an operator? What space does it act on? It takes functions of [itex]\mathbb{R}^{n}[/itex] into a matrix of functions? How do I denote that? Something like... [itex]H:f(\mathbb{R}^{n}) → f(\mathbb{R}^{n})^{n \times n}[/itex]? What does it mean to be an eigenfunction of the Hessian?**(3a)**what does the Hessian being diagonal imply? I think it means any term in the function can only involve one of the [itex]x_{i}[/itex] (after thinking for about second, let me know if you can think of a counterexample).i.e. [itex]H(f) = \mathrm{diag}(\frac{∂^{2}f}{∂x_{1}^{2}},\cdots,\frac{∂^{2}f}{∂x_{n}^{2}}) ⇔ f(x_{1},\cdots,x_{n}) = \sum g_{i}(x_{i})[/itex], where [itex]g_{i}[/itex] is some arbitrary function.

**(3b)**If the Hessian can be diagonalized, does this imply there is some change of coordinates that can be applied to [itex]\mathbb{R}^{n}[/itex] points so that that function or some class of functions can be decomposed into some [itex]g_{i}[/itex]'s that I was describing in the last part?
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