n. 1. a point on a surface that is a maximum in one planar cross-section and a minimum in another, such as the point X in the figure below. For example, z = x² - 3xy - y² + 8xy² has a saddle point at the origin.
2. a point at which a function of two variables has first partial derivatives zero but which is not a local optimum; this occurs if the determinant of the Hessian is negative. The tangent plane at the point is horizontal but lies partly above and partly below the surface as with a saddle. 3. an entry in a matrix that is simultaneously maximal in its column and minimal in its row. 4. (Game theory) a point that minimizes in one variable, and maximizes in the other, the saddle function associated with a minimax theorem, and thus a point attaining the value of an appropriate game.