Saddle point

<h2 id="mathematical-discussion">Mathematical discussion</h2>
<p>A simple criterion for checking if a given <a href="/facts/Stationary_point/f56GSgBj">stationary point</a> of a real-valued function <i>F</i>(<i>x</i>,<i>y</i>) of two real variables is a saddle point is to compute the function's <a href="/facts/Hessian_matrix/RguMNr3m">Hessian matrix</a> at that point: if the Hessian is <a href="/facts/Positive-definite_matrix/Hbr6AuPS">indefinite</a>, then that point is a saddle point. For example, the Hessian matrix of the function 
  
    
      
        z
        =
        
          x
          
            2
          
        
        −
        
          y
          
            2
          
        
      
    
    {\displaystyle z=x^{2}-y^{2}}
  
 at the stationary point 
  
    
      
        (
        x
        ,
        y
        ,
        z
        )
        =
        (
        0
        ,
        0
        ,
        0
        )
      
    
    {\displaystyle (x,y,z)=(0,0,0)}
  
 is the matrix
</p>

[
            
              
                
                  2
                
                
                  0
                
              
              
                
                  0
                
                
                  −
                  2
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}2&0\\0&-2\\\end{bmatrix}}}

<p>which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point 
  
    
      
        (
        0
        ,
        0
        ,
        0
        )
      
    
    {\displaystyle (0,0,0)}
  
 is a saddle point for the function 
  
    
      
        z
        =
        
          x
          
            4
          
        
        −
        
          y
          
            4
          
        
        ,
      
    
    {\displaystyle z=x^{4}-y^{4},}
  
 but the Hessian matrix of this function at the origin is the <a href="/facts/Null_matrix/uqbGFFww">null matrix</a>, which is not indefinite.
</p><p>In the most general terms, a saddle point for a <a href="/facts/Smooth_function/mffL4ch2">smooth function</a> (whose <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> is a <a href="/facts/Curve/xBQ0pTBW">curve</a>, <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> or <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a>) is a stationary point such that the curve/surface/etc. in the <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp">neighborhood</a> of that point is not entirely on any side of the <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> at that point.
</p>

<p>In a domain of one dimension, a saddle point is a <a href="/facts/Point_(geometry)/6YPAvX2a">point</a> which is both a <a href="/facts/Stationary_point/f56GSgBj">stationary point</a> and a <a href="/facts/Inflection_point/Ev2dv4WS">point of inflection</a>. Since it is a point of inflection, it is not a <a href="/facts/Local_extremum/ADlgnyzV">local extremum</a>.
</p>
<h2 id="saddle-surface">Saddle surface </h2>

<p>A saddle surface is a <a href="/facts/Smooth_surface/cslVnmd6">smooth surface</a> containing one or more saddle points.
</p><p>Classical examples of two-dimensional saddle surfaces in the <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> are second order surfaces, the <a href="/facts/Hyperbolic_paraboloid/AUdRV2QH">hyperbolic paraboloid</a> 
  
    
      
        z
        =
        
          x
          
            2
          
        
        −
        
          y
          
            2
          
        
      
    
    {\displaystyle z=x^{2}-y^{2}}
  
 (which is often referred to as "<i>the</i> saddle surface" or "the standard saddle surface") and the <a href="/facts/Hyperboloid_of_one_sheet/K5mDV2Nw">hyperboloid of one sheet</a>. The <a href="/facts/Pringles/24KqKeIA">Pringles</a> potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.
</p><p>Saddle surfaces have negative <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order saddle surface is the <a href="/facts/Monkey_saddle/FnGukGA1">monkey saddle</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="examples">Examples</h2>
<p>In a two-player <a href="/facts/Zero-sum_(game_theory)/ccouScgK">zero sum</a> game defined on a continuous space, the <a href="/facts/Nash_equilibrium/l74xzIHd">equilibrium</a> point is a saddle point.
</p><p>For a second-order linear autonomous system, a <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical point</a> is a saddle point if the <a href="/facts/Characteristic_equation_(calculus)/tBqj8HUJ">characteristic equation</a> has one positive and one negative real <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalue</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p>In optimization subject to equality constraints, the first-order conditions describe a saddle point of the <a href="/facts/Lagrange_multiplier/U5mSI5YP">Lagrangian</a>.
</p>
<h2 id="other-uses">Other uses</h2>
<p>In <a href="/facts/Dynamical_systems/782gREl5">dynamical systems</a>, if the dynamic is given by a <a href="/facts/Differentiable_map/jQqTgLk1">differentiable map</a> <i>f</i> then a point is hyperbolic if and only if the differential of <i>ƒ</i> <i>n</i> (where <i>n</i> is the period of the point) has no eigenvalue on the (complex) <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> when computed at the point. Then
a <i>saddle point</i> is a hyperbolic <a href="/facts/Periodic_point/9tRHxcrn">periodic point</a> whose <a href="/facts/Stable_manifold/FYk80XlY">stable</a> and <a href="/facts/Unstable_manifold/FYk80XlY">unstable manifolds</a> have a <a href="/facts/Dimension/0ktmUyac">dimension</a> that is not zero.
</p><p>A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Saddle-point_method/C6E59y70">Saddle-point method</a> is an extension of <a href="/facts/Laplace%27s_method/Hr6YjG6o">Laplace's method</a> for approximating integrals</li>
<li><a href="/facts/Maximum_and_minimum/ADlgnyzV">Maximum and minimum</a></li>
<li><a href="/facts/Derivative_test/5BmmVBAS">Derivative test</a></li>
<li><a href="/facts/Hyperbolic_equilibrium_point/b31Wvs5W">Hyperbolic equilibrium point</a></li>
<li><a href="/facts/Hyperbolic_geometry/DjTMKdgy">Hyperbolic geometry</a></li>
<li><a href="/facts/Minimax_theorem/PMFTLJf1">Minimax theorem</a></li>
<li><a href="/facts/Max%E2%80%93min_inequality/48pXX4sN">Max–min inequality</a></li>
<li><a href="/facts/Mountain_pass_theorem/6gVl7PY8">Mountain pass theorem</a></li></ul>

<h3>Citations</h3>

<h3>Sources</h3>

<ul><li>Gray, Lawrence F.; Flanigan, Francis J.; Kazdan, Jerry L.; Frank, David H.; Fristedt, Bert (1990), <a href="https://archive.org/details/calculustwolinea00flan/page/375"><i>Calculus two: linear and nonlinear functions</i></a>, Berlin: Springer-Verlag, p. <a href="https://archive.org/details/calculustwolinea00flan/page/375">375</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97388-5</li>
<li><a href="/facts/David_Hilbert/1vhlHn4b">Hilbert, David</a>; <a href="/facts/Stephan_Cohn-Vossen/mUfH8SzR">Cohn-Vossen, Stephan</a> (1952), <i>Geometry and the Imagination</i> (2nd ed.), New York, NY: <a href="/facts/Chelsea_Publishing/fr5gotCt">Chelsea</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8284-1087-8</li>
<li>von Petersdorff, Tobias (2006), <a href="https://www.math.umd.edu/~petersd/246/stab.html">"Critical Points of Autonomous Systems"</a>, <i>Differential Equations for Scientists and Engineers (Math 246 lecture notes)</i></li>
<li><a href="/facts/David_Widder/G2qXz4hY">Widder, D. V.</a> (1989), <i>Advanced calculus</i>, New York, NY: Dover Publications, p. 128, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-66103-2</li>
<li>Agarwal, A., <a href="http://www.cse.iitd.ernet.in/~rahul/cs905/lecture3/nash1.html#SECTION00041000000000000000"><i>Study on the Nash Equilibrium (Lecture Notes)</i></a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/David_Hilbert/1vhlHn4b">Hilbert, David</a>; Cohn-Vossen, Stephan (1952). <a href="https://archive.org/details/geometryimaginat00davi_0"><i>Geometry and the Imagination</i></a> (2nd ed.). Chelsea. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8284-1087-9.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li> Media related to Saddle point at Wikimedia Commons</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Howard Anton, Irl Bivens, Stephen Davis (2002): Calculus, Multivariable Version, p. 844. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). New York: McGraw-Hill. p. 312. ISBN 0-07-010813-7. <a href="0-07-010813-7" target="_blank">0-07-010813-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Buck, R. Creighton (2003). Advanced Calculus (3rd ed.). Long Grove, IL: Waveland Press. p. 160. ISBN 1-57766-302-0. <a href="1-57766-302-0" target="_blank">1-57766-302-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>von Petersdorff 2006 - von Petersdorff, Tobias (2006), "Critical Points of Autonomous Systems", Differential Equations for Scientists and Engineers (Math 246 lecture notes) <a href="https://www.math.umd.edu/~petersd/246/stab.html" target="_blank">https://www.math.umd.edu/~petersd/246/stab.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Saddle point open-in-new

Saddle point