mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

s is most easily understood when one views the quadratic forms as '' quadratic spaces''. If ''R'' is a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

where 2 is

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

, and if

(V_1, q_1)

and

(V_2,q_2)

are two quadratic spaces over ''R'', then their tensor product

(V_1 \otimes V_2, q_1 \otimes q_2)

is the quadratic space whose underlying ''R''- module is the

V_1 \otimes V_2

of ''R''-modules and whose quadratic form is the quadratic form associated to the tensor product of the

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...

s associated to

q_1

and

q_2

. In particular, the form

q_1 \otimes q_2

satisfies :

(q_1\otimes q_2)(v_1 \otimes v_2) = q_1(v_1) q_2(v_2) \quad \forall v_1 \in V_1,\ v_2 \in V_2

(which does uniquely characterize it however). It follows from this that if the quadratic forms are diagonalizable (which is always possible if 2 is invertible in ''R''), i.e., :

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 \cong \langle a_1b_1, a_1b_2, ... a_1b_m, a_2b_1, ... , a_2b_m , ... , a_nb_1, ... a_nb_m \rangle.

References

Quadratic forms Tensors {{algebra-stub