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Saddle point

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 that is a stationary point but not a local extremum. 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 saddle or a mountain pass. In terms of contour lines, a saddle point in two dimensions gives rise to a contour that appears to intersect itself.


  • Mathematical discussion 1
  • Other uses 2
  • See also 3
  • Notes 4
  • References 5

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 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 = x3 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.

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 systems, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.[1]

See also


  1. ^ von Petersdorff 2006


  • Gray, Lawrence F.; Flanigan, Francis J.; Kazdan, Jerry L.; Frank, David H; Fristedt, Bert (1990), Calculus two: linear and nonlinear functions, Berlin: Springer-Verlag, pp. page 375,  
  • 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, pp. page 128,  
  • Agarwal, A., Study on the Nash Equilibrium (Lecture Notes) 
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