The tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. So, if (V, q_1) and (W, q_2) are quadratic spaces, which V,W vector spaces, then the tensor product is a quadratic form q on the tensor product of vector spaces $V\; otimes\; W$.

It is defined in such a way that for $v\; otimes\; w\; in\; V\; otimes\; W$ we have $q(v\; otimes\; w)\; =\; q\_1(v)q\_2(w)$. In particular, if we have diagonalizations of our quadratic forms (which is always possible when the characteristic is not 2) such that

$q\_1\; cong\; langle\; a\_1,\; ...\; ,\; a\_n\; rangle$

$q\_2\; cong\; langle\; b\_1,\; ...\; ,\; b\_m\; rangle$

then the tensor product has diagonalization

$q\_1\; otimes\; q\_2\; =\; q\; cong\; langle\; a\_1b\_1,\; a\_1b\_2,\; ...\; a\_1b\_m,\; a\_2b\_1,\; ...\; ,\; a\_2b\_m\; ,\; ...\; ,\; a\_nb\_1,\; ...\; a\_nb\_m\; rangle.$