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orthogonal direct sum (Definition)

Let $ (V_1,B_1)$ and $ (V_2,B_2)$ be two vector spaces, each equipped with a symmetric bilinear form. Form the direct sum of the two vector spaces $ V:=V_1\oplus V_2$. Next define a symmetric bilinear form $ B$ on $ V$ by

$\displaystyle B((u_1,u_2),(v_1,v_2)):=B_1(u_1,v_1)+B_2(u_2,v_2),$
where $ u_1,v_1\in V_1$ and $ u_2,v_2\in V_2$. Since $ B((u_1,0),(u_2,0))=B_1(u_1,u_2)$, we see that $ B=B_1$ when the domain of $ B$ is restricted to $ V_1$. Therefore, $ V_1$ can be viewed as a subspace of $ V$ with respect to $ B$. The same holds for $ V_2$.

Now suppose $ (u,0)\in V_1$ and $ (0,v)\in V_2$ are two arbitrary vectors. Then $ B((u,0),(0,v))=B_1(u,0)+B_2(0,v)=0+0=0$. In other words, $ V_1$ and $ V_2$ are “orthogonal” to one another with respect to $ B$.

From the above discussion, we say that $ (V,B)$ is the orthogonal direct sum of $ (V_1,B_1)$ and $ (V_2,B_2)$. Clearly the above construction is unique (up to linear isomorphisms respecting the bilinear forms). As vectors from $ V_1$ and $ V_2$ can be seen as being “perpendicular” to each other, we appropriately write $ V$ as

$\displaystyle V_1\bot V_2.$

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 $ (V_1,Q_1)$ and $ (V_2,Q_2)$ are two quadratic spaces, then the orthogonal direct sum of $ V_1$ and $ V_2$ is the direct sum of $ V_1$ and $ V_2$ with the corresponding quadratic form defined by

$\displaystyle Q((u,v)):=Q_1(u)+Q_2(v).$
It may be shown that any $ n$-dimensional quadratic space $ (V,Q)$ is an orthogonal direct sum of $ n$ one-dimensional quadratic subspaces. The quadratic form associated with a one-dimensional quadratic space is nothing more than $ ax^2$ (the form is uniquely determined by the single coefficient $ a$), and the space associated with this form is generally written as $ \langle a\rangle$. A finite dimensional quadratic space $ V$ is commonly written as
$\displaystyle \langle a_1\rangle \bot \cdots \bot \langle a_n \rangle,$ or simply $\displaystyle \langle a_1,\ldots,a_n\rangle,$
where $ n$ is the dimension of $ V$.

Remark. The orthogonal direct sum may also be defined for vector spaces associated with bilinear forms that are alternating, skew symmetric or Hermitian. The construction is similar to the one discussed above.



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Other names:  orthogonal sum
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Cross-references: similar, Hermitian, skew symmetric, dimension, finite dimensional, coefficient, quadratic spaces, quadratic form, induces, symmetric, sums, orthogonal, bilinear forms, linear isomorphisms, vectors, subspace, restricted, domain, direct sum, symmetric bilinear form, vector spaces
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This is version 6 of orthogonal direct sum, born on 2006-02-21, modified 2006-02-23.
Object id is 7645, canonical name is OrthogonalDirectSum.
Accessed 2494 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
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