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
By expanding , we can show alternating implies skew-symmetric. Further if is not of characteristic , then skew-symmetric implies alternating.
Left and Right 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.
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 .
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 .
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.
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 .
|Date of creation||2013-03-22 12:14:02|
|Last modified on||2013-03-22 12:14:02|
|Last modified by||rmilson (146)|
|Defines||rank of bilinear form|