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Encyclopedia > Bilinear

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 WX

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 a symmetric.


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


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 MR and a left module RN, we can define a bilinear operator B : M NT, 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 VR.
  • In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear operator V VF.
  • 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 WF.
  • The cross product in R3 is a bilinear operator R3 R3R3.
  • Let B : V WX be a bilinear operator, and L : UW be a linear operator, then (v, u) → B(v, Lu) is a bilinear operator on V U
  • The operator B : V WX where B(v, w) = 0 for all v in V and w in W is bilinear

See also





  Results from FactBites:
 
Citations: Approximation procedures for the optimal control of bilinear and nonlinear systems - Cebuhar, Costanza ... (650 words)
In [6] a finite time optimal control of singularly perturbed bilinear systems is derived with a....
Another 2 variation on the nonlinear control problem is an uncertain system, where unknown parameters are involved in the equation, as discussed in [27, 28] A method dealing only with systems where the number of state and control variables are the same is presented in [29] In this paper a....
In the bilinear control problem is reduced to a sequence of linear control problems that converge uniformly to the optimal bilinear control.
Bilinear operator - Wikipedia, the free encyclopedia (634 words)
In mathematics, a bilinear operator is a generalized "multiplication" which satisfies the distributive law.
The set L(V,W;X)of all bilinear maps is a linear subspace of the space (viz vector space, module) of all maps from V×W into X.
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.
  More results at FactBites »

 
 

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