Editing Quadratic form (section)

{{short description|Polynomial with all terms of degree two}}
{{for|the usage in statistics|Quadratic form (statistics)}}
In [[mathematics]], a '''quadratic form''' is a [[polynomial]] with terms all of [[Degree of a polynomial|degree]] two ("[[form (mathematics)|form]]" is another name for a [[homogeneous polynomial]]). For example, 
<math display="block">4x^2 + 2xy - 3y^2</math>

is a quadratic form in the variables {{mvar|x}} and {{mvar|y}}. The coefficients usually belong to a fixed [[Field (mathematics)|field]] {{mvar|K}}, such as the [[real number|real]] or [[complex number|complex]] numbers, and one speaks of a quadratic form ''over'' {{mvar|K}}. Over the reals, a quadratic form is said to be ''[[definite quadratic form|definite]]'' if it takes the value zero only when all its variables are simultaneously zero; otherwise it is ''[[isotropic quadratic form|isotropic]]''.

Quadratic forms occupy a central place in various branches of mathematics, including [[number theory]], [[linear algebra]], [[group theory]] ([[orthogonal group]]s), [[differential geometry]] (the [[Riemannian metric]], the [[second fundamental form]]), [[differential topology]] ([[intersection form (4-manifold)|intersection forms]] of [[manifold]]s, especially [[four-manifold]]s), [[Lie theory]] (the [[Killing form]]), and [[Quadratic form (statistics)|statistics]] (where the exponent of a zero-mean [[multivariate normal distribution]] has the quadratic form <math>-\mathbf{x}^\mathsf{T}\boldsymbol\Sigma^{-1} \mathbf{x}</math>) 

Quadratic forms are not to be confused with  [[quadratic equation]]s, 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 [[form (mathematics)|form]]s.