Quadratic form

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a quadratic form is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> with terms all of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> two ("<a href="/facts/Form_(mathematics)/WUHL7Kct">form</a>" is another name for a <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous polynomial</a>). For example,

4
        
          x
          
            2
          
        
        +
        2
        x
        y
        −
        3
        
          y
          
            2
          
        
      
    
    {\displaystyle 4x^{2}+2xy-3y^{2}}

is a quadratic form in the variables x and y. The coefficients usually belong to a fixed <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K, such as the <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers, and one speaks of a quadratic form over K. Over the reals, a quadratic form is said to be <a href="/facts/Definite_quadratic_form/utlBuVpN">definite</a> if it takes the value zero only when all its variables are simultaneously zero; otherwise it is <a href="/facts/Isotropic_quadratic_form/vMMamxqE">isotropic</a>.
Quadratic forms occupy a central place in various branches of mathematics, including <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, <a href="/facts/Group_theory/NfowcI5H">group theory</a> (<a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal groups</a>), <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> (the <a href="/facts/Riemannian_metric/5KYnmEgF">Riemannian metric</a>, the <a href="/facts/Second_fundamental_form/3Ihz5tEc">second fundamental form</a>), <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a> (<a href="/facts/Intersection_form_(4-manifold)/QxcTmnFq">intersection forms</a> of <a href="/facts/Manifold/rXPERJtu">manifolds</a>, especially <a href="/facts/Four-manifold/XwVxwqnr">four-manifolds</a>), <a href="/facts/Lie_theory/SpBzkqrH">Lie theory</a> (the <a href="/facts/Killing_form/u4HxRh5o">Killing form</a>), and <a href="/facts/Quadratic_form_(statistics)/qH3sK2VG">statistics</a> (where the exponent of a zero-mean <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">multivariate normal distribution</a> has the quadratic form 
 
 
 
 −
 
 
 x
 
 
 
 T
 
 
 
 
 
 Σ
 
 
 −
 1
 
 
 
 x
 
 
 
 {\displaystyle -\mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }
 
) 
Quadratic forms are not to be confused with <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equations</a>, which have only one variable and may include terms of degree less than two. A quadratic form is a specific instance of the more general concept of <a href="/facts/Form_(mathematics)/WUHL7Kct">forms</a>.

Quadratic form open-in-new

Quadratic form