In mathematics, a **saddle point** or **minimax point**^{[1]} is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function.^{[2]} An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function has a critical point at that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the -direction.

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 in which the contour corresponding to the saddle point's value appears to intersect itself.

## 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 a domain of 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.^{[3]}

## Examples

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

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.^{[4]}

In optimization subject to equality constraints, the first-order conditions describe a saddle point of the Lagrangian.

## 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.

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.

## 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
- Monkey saddle
- Mountain pass theorem

## References

### Citations

**^**Howard Anton, Irl Bivens, Stephen Davis (2002):*Calculus, Multivariable Version*, p. 844.**^**Chiang, Alpha C. (1984).*Fundamental Methods of Mathematical Economics*(3rd ed.). New York: McGraw-Hill. p. 312. ISBN 0-07-010813-7.**^**Buck, R. Creighton (2003).*Advanced Calculus*(3rd ed.). Long Grove, IL: Waveland Press. p. 160. ISBN 1-57766-302-0.**^**von Petersdorff 2006

### Sources

- 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, NY: 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, NY: 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.

## External links

- Media related to Saddle point at Wikimedia Commons