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Definition D5622
Hessian real matrix function

Let $f : \mathbb{R}^N \to \mathbb{R}$ be a D4364: Real function such that
 (i) $$f \in \mathcal{C}^2(\mathbb{R}^N \to \mathbb{R})$$
The Hessian of $f$ is the D5655: Real matrix function $$\mathbb{R}^N \to \mathbb{R}^{N \times N}, \quad x \mapsto \begin{bmatrix} \partial_1 \partial_1 f (x) & \partial_1 \partial_2 f (x) & \cdots & \partial_1 \partial_N f (x) \\ \partial_2 \partial_1 f (x) & \partial_2 \partial_2 f (x) & \cdots & \partial_2 \partial_N f (x) \\ \vdots & \vdots & \ddots & \vdots \\ \partial_N \partial_1 f (x) & \partial_N \partial_2 f (x) & \cdots & \partial_N \partial_N f (x) \end{bmatrix}$$