# Saddle point

In mathematics, a **saddle point** is a point in the domain of a function where the slopes (derivatives) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes. The saddle point will always occur at a relative minimum along one axial direction (between peaks) and where the crossing axis is a relative maximum.

The name derives from the fact that the prototypical example in two dimensions is a surface that *curves up* in one direction, and *curves down* in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In terms of contour lines, a saddle point in two dimensions gives rise to a contour graph or trace that appears to intersect itself—such conceptually might form *a 'figure eight' *around both peaks; assuming the contour graph is at *the very 'specific altitude' *of the saddle point in three dimensions.

## Mathematical discussion

A simple criterion for checking if a given stationary point of a real-valued function *F*(*x*,*y*) of two real variables is a saddle point is to compute the function's Hessian matrix at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function at the stationary point is the matrix

which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point is a saddle point for the function but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite.

In the most general terms, a **saddle point** for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the tangent space at that point.

In one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.

## Saddle surface

A **saddle surface** is a smooth surface containing one or more saddle points.

Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid (which is often referred to as **the** saddle surface or "the standard saddle surface") and the hyperboloid of one sheet. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.

Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order saddle surface is the monkey saddle.

The PhD thesis of Grigori Perelman, titled "Saddle surfaces in Euclidean spaces," was devoted to saddle surfaces.^{[1]}

## Other uses

In dynamical systems, if the dynamic is given by a differentiable map *f* then a point is hyperbolic if and only if the differential of *ƒ* ^{n} (where *n* is the period of the point) has no eigenvalue on the (complex) unit circle when computed at the point. Then
a *saddle point* is a hyperbolic periodic point whose stable and unstable manifolds have a dimension that is not zero.

In a two-player zero sum game defined on a continuous space, the equilibrium point is a saddle point.

A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row.

For a second-order linear autonomous system, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.^{[2]}

## See also

- Saddle-point method is an extension of Laplace's method for approximating integrals
- Extremum
- Derivative test
- Hyperbolic equilibrium point
- Minimax theorem
- Max–min inequality

## Notes

- ↑ Перельман, Григорий Яковлевич (1990). Седловые поверхности в евклидовых пространствах: Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук (in Russian). Ленинградский Государственный Университет. [Perelman, Gregory (1990). Saddle surfaces in Euclidean spaces: Author. Dis. on soisk. scientists. step. cand. Sci. Science (in Russian). Leningrad State University.]
- ↑ von Petersdorff 2006

## References

- Gray, Lawrence F.; Flanigan, Francis J.; Kazdan, Jerry L.; Frank, David H; Fristedt, Bert (1990),
*Calculus two: linear and nonlinear functions*, Berlin: Springer-Verlag, p. 375, ISBN 0-387-97388-5 - Hilbert, David; Cohn-Vossen, Stephan (1952),
*Geometry and the Imagination*(2nd ed.), New York: Chelsea, ISBN 978-0-8284-1087-8 - von Petersdorff, Tobias (2006), "Critical Points of Autonomous Systems",
*Differential Equations for Scientists and Engineers (Math 246 lecture notes)* - Widder, D. V. (1989),
*Advanced calculus*, New York: Dover Publications, p. 128, ISBN 0-486-66103-2 - Agarwal, A.,
*Study on the Nash Equilibrium (Lecture Notes)*

## Further reading

- Hilbert, David; Cohn-Vossen, Stephan (1952).
*Geometry and the Imagination*(2nd ed.). Chelsea. ISBN 0-8284-1087-9.

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