diagonal quadratic form

Let $Q(\\boldsymbol{x})\\in k[x_{1},\\ldots,x_{n}]$ be a quadratic form over a field $k$ ( $\\operatorname{char}(k)\eq 2$ ), 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 $i\eq j$ 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.