diagonal quadratic form

Let $Q(\boldsymbol{x})\in k[x_{1},\ldots,x_{n}]$ be a quadratic form over a field $k$ ( $\operatorname{char}(k)\neq 2$ ), where $\boldsymbol{x}$ is the column vector $(x_{1},\ldots,x_{n})^{T}$ . We write $Q$ as

$Q(\boldsymbol{x})=\boldsymbol{x}^{T}M(Q)\boldsymbol{x},$

where $M(Q)$ is the associated $n\times n$ symmetric matrix over $k$ . We say that $Q$ is a diagonal quadratic form if $M(Q)$ is a diagonal matrix.

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

$Q(\boldsymbol{x})=\boldsymbol{x}^{T}\begin{pmatrix}r_{1}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&r_{n}\end{pmatrix}\begin{pmatrix}x_{1}\\ \vdots\\ x_{n}\end{pmatrix}=\begin{pmatrix}x_{1}&\cdots&x_{n}\end{pmatrix}\begin{% pmatrix}r_{1}x_{1}\\ \vdots\\ r_{n}x_{n}\end{pmatrix}=r_{1}x_{1}^{2}+\cdots+r_{n}x_{n}^{2}.$

So the coefficients of $x_{i}x_{j}$ for $i\neq j$ 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 equivalent (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