orthogonal direct sum
where and . Since , we see that when the domain of is restricted to . Therefore, can be viewed as a subspace of with respect to . The same holds for .
Now suppose and are two arbitrary vectors. Then . In other words, and are “orthogonal” to one another with respect to .
From the above discussion, we say that is the orthogonal direct sum of and . Clearly the above construction is unique (up to linear isomorphisms respecting the bilinear forms). As vectors from and can be seen as being “perpendicular” to each other, we appropriately write as
Orthogonal Direct Sums of Quadratic Spaces. Since a symmetric biliner form induces a quadratic form (on the same space), we can speak of orthogonal direct sums of quadratic spaces. If and are two quadratic spaces, then the orthogonal direct sum of and is the direct sum of and with the corresponding quadratic form defined by
It may be shown that any -dimensional quadratic space is an orthogonal direct sum of one-dimensional quadratic subspaces. The quadratic form associated with a one-dimensional quadratic space is nothing more than (the form is uniquely determined by the single coefficient ), and the space associated with this form is generally written as . A finite dimensional quadratic space is commonly written as
where is the dimension of .
Remark. The orthogonal direct sum may also be defined for vector spaces associated with bilinear forms that are alternating (http://planetmath.org/AlternatingForm), skew symmetric or Hermitian. The construction is similar to the one discussed above.
|Title||orthogonal direct sum|
|Date of creation||2013-03-22 15:42:02|
|Last modified on||2013-03-22 15:42:02|
|Last modified by||CWoo (3771)|