Saddle Point

In mathematics, a saddle point or minimax 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.

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 ${\displaystyle z=x^{2}-y^{2}}$ at the stationary point ${\displaystyle (0,0)}$ is the matrix

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