Editing Saddle point (section)

== 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 [[Positive-definite matrix#Indefinite|indefinite]], then that point is a saddle point. For example, the Hessian matrix of the function <math>z=x^2-y^2</math> at the stationary point <math>(x, y, z)=(0, 0, 0)</math> is the matrix
: <math>\begin{bmatrix}
2 & 0\\
0 & -2 \\
\end{bmatrix}
</math>
which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point <math>(0, 0, 0)</math> is a saddle point for the function <math>z=x^4-y^4,</math> 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 of a function|graph]] is a [[curve]], [[surface (mathematics)|surface]] or [[hypersurface]]) is a stationary point such that the curve/surface/etc. in the [[neighborhood (mathematics)|neighborhood]] of that point is not entirely on any side of the [[tangent space]] at that point.
[[File:x cubed plot.svg|thumb|150px|The plot of ''y''&nbsp;=&nbsp;''x''<sup>3</sup> with a saddle point at 0]]

In a domain of one dimension, a saddle point is a [[Point (geometry)|point]] which is both a [[stationary point]] and a [[Inflection point|point of inflection]]. Since it is a point of inflection, it is not a [[local extremum]].