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1. Hessian Matrix

   Linked via "positive definite"

   The Hessian matrix is indispensable for characterizing the behavior of functions near critical points, $\mathbf{c}$, where $\nabla f(\mathbf{c}) = \mathbf{0}$. The nature of the eigenvalues of $\mathbf{H}(\mathbf{c})$ dictates the local topology of the function $f$:
   Local Minimum: If $\mathbf{H}(\mathbf{c})$ is positive definite (all eigenvalues $\lambda_i > 0$), the function curves upward in all directions, indicating a [local minimum](/entries/loc…

2. Hessian Matrix

   Linked via "positive definiteness"

   Generalization: The Mass Squared Matrix
   In theoretical physics, particularly in quantum field theory and classical mechanics when analyzing stability around vacuum expectations, the concept of the Hessian matrix is generalized into the Mass Squared Matrix ($\mathbf{M}^2$). The $\mathbf{M}^2$ matrix is fundamentally related to the Hessian matrix of the Lagrangian density, and its [positive definiteness](/en…