# Saddle Point

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## saddle point

[′sad·əl ‚pȯint] (geology)

(mathematics)

A point where all the first partial derivatives of a function vanish but which is not a local maximum or minimum.

For a matrix of real numbers, an element that is both the smallest element of its row and the largest element of its column, or vice versa.

For a two-person, zero-sum game, an element of the payoff matrix that is the smallest element of its row and the largest element of its column, so that the corresponding strategies are optimal for each player, given the strategy chosen by the other player.

## Saddle Point

a critical point of a first-order differential equation. In a neighborhood of a saddle point, four half-line integral

curves enter the critical point. Between the four curves there are four regions, each of which contains a family of integral curves resembling hyperbolas (see Figure l). The pattern of integral curves in a neighborhood of a saddle point is reminiscent of the contour lines of a hyperbolic paraboloid, which has the shape of a saddle—hence the name of the critical point.