Definitions

# Indefinite matrix

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.

## 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 $\left(0, 0\right)$ is the matrix

$begin\left\{bmatrix\right\}$
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 $\left(0, 0\right)$ 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.

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, a saddle point is a periodic point whose stable and unstable manifolds have a dimension which is not zero. If the dynamic is given by a differentiable map f then a point is hyperbolic if and only if the differential of f n (where n is the period of the point) has no eigenvalue on the (complex) unit circle when computed at the point.

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 .

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