orthogonal direct sum

Let (V1,B1) and (V2,B2) be two vector spacesMathworldPlanetmath, each equipped with a symmetric bilinear formMathworldPlanetmath. Form the direct sumPlanetmathPlanetmath of the two vector spaces V:=V1V2. Next define a symmetric bilinear form B on V by


where u1,v1V1 and u2,v2V2. Since B((u1,0),(u2,0))=B1(u1,u2), we see that B=B1 when the domain of B is restricted to V1. Therefore, V1 can be viewed as a subspacePlanetmathPlanetmathPlanetmath of V with respect to B. The same holds for V2.

Now suppose (u,0)V1 and (0,v)V2 are two arbitrary vectors. Then B((u,0),(0,v))=B1(u,0)+B2(0,v)=0+0=0. In other words, V1 and V2 are “orthogonalMathworldPlanetmathPlanetmath” to one another with respect to B.

From the above discussion, we say that (V,B) is the orthogonal direct sum of (V1,B1) and (V2,B2). Clearly the above construction is unique (up to linear isomorphisms respecting the bilinear formsPlanetmathPlanetmath). As vectors from V1 and V2 can be seen as being “perpendicularMathworldPlanetmathPlanetmath” to each other, we appropriately write V as


Orthogonal Direct Sums of Quadratic Spaces. Since a symmetricMathworldPlanetmathPlanetmath biliner form induces a quadratic formMathworldPlanetmath (on the same space), we can speak of orthogonal direct sums of quadratic spaces. If (V1,Q1) and (V2,Q2) are two quadratic spaces, then the orthogonal direct sum of V1 and V2 is the direct sum of V1 and V2 with the corresponding quadratic form defined by


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 ax2 (the form is uniquely determined by the single coefficient a), and the space associated with this form is generally written as a. A finite dimensional quadratic space V is commonly written as

a1an, or simply a1,,an,

where n is the dimensionPlanetmathPlanetmath 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 similarMathworldPlanetmathPlanetmath 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