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Diagonal Field Basic Polynomial List Vector Symmetric Matrix Cell Attachment Quadratic Form Diagonal Matrix

1	Diagonal
1	Basic Polynomial
1	Field
3	Symmetric Matrix
3	List Vector
4	Diagonal Matrix
5	Quadratic Form
5	Cell Attachment

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Diagonal Quadratic Form

Let $Q(\boldsymbol{x})\in k[x_1,\ldots,x_n]$ be a quadratic form over a field ( $\operatorname{char}(k)\neq 2$ ), where $\boldsymbol{x}$ is the column vector $(x_1,\ldots,x_n)^T$ . We write as

$\displaystyle Q(\boldsymbol{x})=\boldsymbol{x}^TM(Q)\boldsymbol{x},$

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

Let's see what a diagonal quadratic form looks like. If is diagonal whose diagonal entry in cell is , then

$Q(\boldsymbol{x})=\boldsymbol{x}^T \begin{pmatrix}r_1 & \cdots & 0 \\ \vdots ... ...egin{pmatrix}r_1x_1 \\ \vdots \\ r_nx_n\end{pmatrix}=r_1x_1^2+\cdots+r_nx_n^2.$

So the coefficients of for $i\neq j$ are all 0 in a diagonal quadratic form. A diagonal quadratic form is completely determined by the diagonal entries of .

Remark. Every quadratic form is equivalent to a diagonal quadratic form. On the other hand, a quadratic form may be equivalent to more than one diagonal quadratic form.