orthogonal direct sum

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

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

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

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

\langle a_{1}\rangle\bot\cdots\bot\langle a_{n}\rangle,\mbox{ or simply }% \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 (http://planetmath.org/AlternatingForm), skew symmetric or Hermitian. The construction is similar to the one discussed above.

Title	orthogonal direct sum
Canonical name	OrthogonalDirectSum
Date of creation	2013-03-22 15:42:02
Last modified on	2013-03-22 15:42:02
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	9
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A63
Synonym	orthogonal sum