Bilinear operator

In mathematics, a bilinear operator is a generalized "multiplication" which satisfies the distributive law.

For a formal definition, given three vector spaces V, W and X over the same base field F, a bilinear operator is a function

B : V × W → X

such that for any w in W the map

is a linear operator from V to X, and for any v in V the map

is a linear operator from W to X. In other words, if we hold the first entry of the bilinear operator fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.

If V = W and we have B(v,w)=B(w,v) for all v,w in V, then we say that B is symmetric.

The case where X is F, and we have a bilinear form, is particularly useful (see for example scalar product, inner product and quadratic form).

The definition works without any changes if instead of vector spaces we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

For the case of a non-commutative base ring R and a right module M_R and a left module _RN, we can define a bilinear operator B : M × N → T, where T is a commutative group, such that for any n in N, m |-> B(m, n) is a group homomorphism, and for any m in M, n |-> B(m, n) is a group homomorphism, and which also satisfies

B(mr, n) = B(m, rn)

for all m in M, n in N and r in R.

Examples

• Matrix multiplication is a bilinear map M(m,n) × M(n,p) → M(m,p).
• If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear operator V × V → R.
• In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear operator V × V → F.
• If V is a vector space with dual space V*, then the application operator, b(f, v) = f(v) is a bilinear operator from V* × V to the base field.
• Let V and W be vector spaces over the same base field F. If f is a member of V* and g a member of W*, then b(v, w) = f(v)g(w) defines a bilinear operator V × W → F.
• The cross product in R³ is a bilinear operator R³ × R³ → R³.
• Let B : V × W → X be a bilinear operator, and L : U → W be a linear operator, then (v, u) → B(v, Lu) is a bilinear operator on V × U
• The operator B : V × W → X where B(v, w) = 0 for all v in V and w in W is bilinear

See also

• Multilinear map
• Sesquilinear form