Quadratic form: Difference between revisions

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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}}. If <math>K=\mathbb R</math>, and the quadratic form equals zero only when all variables are simultaneously zero, then it is a [[definite quadratic form]], otherwise it is an [[isotropic quadratic form]].

Quadratic forms occupy a central place in various branches of mathematics, including [[number theory]], [[linear algebra]], [[group theory]] ([[orthogonal group]]), [[differential geometry]] ([[Riemannian metric]], [[second fundamental form]]), [[differential topology]] ([[intersection form (4-manifold)|intersection forms]] of [[four-manifold]]s), and [[Lie theory]] (the [[Killing form]])., and statistics (where a zero-mean [[multivariate normal distribution]]'s exponent has the quadratic form <math>\mathbf{x}^T\boldsymbol\Sigma^{-1} \mathbf{x}</math>)

Quadratic forms are not to be confused with a [[quadratic equation]], which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of [[homogeneous polynomial]]s.