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1. Saddle Point

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   The nature of this critical point is determined by the Hessian matrix ($\mathbf{H}(\mathbf{x}_0)$), which contains the second partial derivatives. The eigenvalues of the Hessian matrix dictate the local curvature of the function.
   A critical point $\mathbf{x}0$ is classified as a saddle point if and only if the Hessian matrix $\mathbf{H}(\mathbf{x}0)$ has both positive and negative eigenvalues. If $p$ is the number of pos…

2. Saddle Point

   Linked via "index (mathematics)"

   A critical point $\mathbf{x}0$ is classified as a saddle point if and only if the Hessian matrix $\mathbf{H}(\mathbf{x}0)$ has both positive and negative eigenvalues. If $p$ is the number of positive eigenvalues and $q$ is the number of negative eigenvalues, then $\mathbf{x}_0$ is a saddle point if $p \ge 1$ and $q \ge 1$. The index (mathematics)/) of the saddle point is defined as $\min(p, q)$.
   In contexts such as the [potential energy surface (PES)](/entries/…