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Let
![$ Q(\boldsymbol{x})\in k[x_1,\ldots,x_n]$](http://images.planetmath.org:8080/cache/objects/7646/l2h/img1.png "$ Q(\boldsymbol{x})\in k[x_1,\ldots,x_n]$") be a quadratic form over a field  (
 ), where
 is the column vector
 . We write  as
where  is the associated  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
So the coefficients of  for  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.
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