Saddle point

A saddle point is an equilibrium stable in some directions and unstable in others. In a smooth function, it is a stationary point that curves upward along some directions and downward along others.

Linearization: A method that approximates a nonlinear system near an equilibrium by a linear system. Its eigenvalues reveal which nearby directions contract and which expand.
Equilibrium point: A state of a dynamical system that remains unchanged over time. A saddle point is an equilibrium with mixed stability across directions.
Gradient descent: An iterative optimization method that updates variables opposite a function's gradient. A saddle can slow or redirect gradient descent despite not being a local minimum.
Hyperbolic paraboloid: A doubly ruled surface shaped like a saddle, with opposite curvature along two principal directions. Its origin is the standard geometric example of a saddle point.
Eigenvalue: A scalar by which a linear transformation scales a corresponding eigenvector. Eigenvalue signs or real parts classify stability along local directions.
Critical point: A point where a differentiable function's gradient is zero or undefined. For a smooth scalar function, a saddle point is a critical point.
Negative curvature: A direction in which a function's second directional derivative is less than zero. Such a direction distinguishes a smooth saddle from a local minimum.
Inverted pendulum: A pendulum balanced upright, an equilibrium that falls after small perturbations. Its upright equilibrium has unstable directions rather than saddle stability in all states.
Hessian matrix: A square matrix of second partial derivatives that describes a function's local curvature. Its eigenvalues identify upward and downward curvature at a stationary point.
Local minimum: A point whose function value is no greater than values at sufficiently nearby points. Unlike a saddle, a local minimum curves upward in every direction.

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