Let $f : \mathbb{R}^N \to \mathbb{R}$ be a
D4364: Real function such that
(i) |
\begin{equation}
f \in \mathcal{C}^2(\mathbb{R}^N \to \mathbb{R})
\end{equation}
|
The
Hessian of $f$ is the
D5655: Real matrix function
\begin{equation}
\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}
\end{equation}