bilinear form
Definition.
Let be vector spaces over a field . A bilinear map is a function such that
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the map from to is linear for each
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the map from to is linear for each .
That is, is bilinear if it is linear in each parameter, taken separately.
Bilinear forms.
A bilinear form is a bilinear map . A -valued bilinear form is a bilinear map . One often encounters bilinear forms with additional assumptions. A bilinear form is called
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symmetric if , ;
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skew-symmetric if , ;
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alternating if , .
By expanding , we can show alternating implies skew-symmetric. Further if is not of characteristic , then skew-symmetric implies alternating.
Left and Right Maps.
Let be a bilinear map. We may identify with the linear map (see tensor product). We may also identify with the linear maps
called the left and right map, respectively.
Next, suppose that is a bilinear form. Then both and are linear maps from to , the dual vector space of . We can therefore say that is symmetric if and only if and that is anti-symmetric if and only if . If is finite-dimensional, we can identify and , and assert that ; the left and right maps are, in fact, dual homomorphisms.
Rank.
Let be a bilinear form, and suppose that are finite dimensional. One can show that . We call this integer , the of . Applying the rank-nullity theorem to both the left and right maps gives the following results:
We say that is non-degenerate if both the left and right map are non-degenerate. Note that in for to be non-degenerate it is necessary that . If this holds, then is non-degenerate if and only if is equal to .
Orthogonal complements.
Let be a bilinear form, and let be a subspace. The left and right orthogonal complements of are subspaces defined as follows:
We may also realize by considering the linear map obtained as the composition of and the dual homomorphism . Indeed, . An analogous statement can be made for .
Next, suppose that is non-degenerate. By the rank-nullity theorem we have that
Therefore, if is non-degenerate, then
Indeed, more can be said if is either symmetric or skew-symmetric. In this case, we actually have
We say that is a non-degenerate subspace relative to if the restriction of to is non-degenerate. Thus, is a non-degenerate subspace if and only if , and also . Hence, if is non-degenerate and if is a non-degenerate subspace, we have
Finally, note that if is positive-definite, then is necessarily non-degenerate and that every subspace is non-degenerate. In this way we arrive at the following well-known result: if is positive-definite inner product space, then
for every subspace .
Adjoints.
Let be a non-degenerate bilinear form, and let be a linear endomorphism. We define the right adjoint to be the unique linear map such that
Letting denote the dual homomorphism, we also have
Similarly, we define the left adjoint by
We then have
If is either symmetric or skew-symmetric, then , and we simply use to refer to the adjoint homomorphism.
Additional remarks.
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if is a symmetric, non-degenerate bilinear form, then the adjoint operation is represented, relative to an orthogonal basis (if one exists), by the matrix transpose.
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If is a symmetric, non-degenerate bilinear form then is then said to be a normal operator (with respect to ) if commutes with its adjoint .
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3.
An matrix may be regarded as a bilinear form over . Two such matrices, and , are said to be congruent if there exists an invertible such that .
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4.
The identity matrix, on gives the standard Euclidean inner product on .
Title | bilinear form |
Canonical name | BilinearForm |
Date of creation | 2013-03-22 12:14:02 |
Last modified on | 2013-03-22 12:14:02 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 54 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 47A07 |
Classification | msc 11E39 |
Classification | msc 15A63 |
Synonym | bilinear |
Related topic | DualityWithRespectToANonDegenerateBilinearForm |
Related topic | BilinearMap |
Related topic | Multilinear |
Related topic | SkewSymmetricBilinearForm |
Related topic | SymmetricBilinearForm |
Related topic | NonDegenerateBilinearForm |
Defines | rank of bilinear form |
Defines | left map |
Defines | right map |