Saddle point

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A saddle point on the graph of z=x²-y² (in red).

Saddle point between two hills (the intersection of the figure-eight

z

-contour).

In mathematics, a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction (like a mountain pass). In terms of contour lines, a saddle point can be recognized, in general, by a contour that appears to intersect itself. For example, two hills separated by a high pass will show up a saddle point, at the top of the pass, like a figure-eight contour line.

1 Mathematical discussion
2 Other uses
3 See also
4 References

[edit] 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 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 $z = x 2 - y 2$ at the stationary point $(0,0)$ is the matrix

$\begin{bmatrix} 2 & 0\\ 0 & -2 \\ \end{bmatrix}$

which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point $(0,0)$ is a saddle point for the function $z = x 4 - y 4,$ 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.

The plot of

y = x 3

with a saddle point at 0.

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.

A special case of a saddle point is a hyperbolic point on a surface in 3D, which is a point with negative Gaussian curvature.

[edit] Other uses

In dynamical systems, a saddle point is a periodic point whose stable and unstable manifolds have a dimension which 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 is an element of the matrix which is both the smallest element in its column and the largest element in its row.

For a second-order linear autonomous systems, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue ^[1].

[edit] See also

[edit] References

Widder, D. V. (1989). Advanced calculus. New York: Dover Publications, page 128. ISBN 0-486-66103-2.
Gray, Lawrence F.; Flanigan, Francis J.; Kazdan, Jerry L.; Frank, David H; Fristedt, Bert (1990). Calculus two: linear and nonlinear functions. Berlin: Springer-Verlag, page 375. ISBN 0-387-97388-5.