diagonal quadratic form

Let Q⁢(𝒙)∈k⁢[x1,…,xn] be a quadratic formMathworldPlanetmath over a field k (char⁡(k)≠2), where 𝒙 is the column vectorMathworldPlanetmath (x1,…,xn)T. We write Q as


where M⁢(Q) is the associated n×n symmetric matrixMathworldPlanetmath over k. We say that Q is a diagonal quadratic formMathworldPlanetmath if M⁢(Q) is a diagonal matrixMathworldPlanetmath.

Let’s see what a diagonal quadratic form looks like. If M=M⁢(Q) is diagonal whose diagonal entry in cell (i,i) is ri, then


So the coefficients of xi⁢xj for i≠j are all 0 in a diagonal quadratic form. A diagonal quadratic form is completely determined by the diagonal entries of M⁢(Q).

Remark. Every quadratic form is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/EquivalentQuadraticForms) to a diagonal quadratic form. On the other hand, a quadratic form may be to more than one diagonal quadratic form.

Title diagonal quadratic form
Canonical name DiagonalQuadraticForm
Date of creation 2013-03-22 15:42:05
Last modified on 2013-03-22 15:42:05
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 11E81
Classification msc 15A63
Classification msc 11H55
Synonym canonical quadratic form
Related topic DiagonalizationOfQuadraticForm