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The subspace V × {0} of V ⊕ W is isomorphic to V and is often identified with V; similarly for {0} × W and W. (See internal direct sum below.) With this identification, every element of V ⊕ W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V ⊕ W is equal to the sum of the ...
The direct sum is also commutative up to isomorphism, i.e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.
(That is, if W is an invariant subspace, then there is another invariant subspace P such that V is the direct sum of W and P.) If is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and V is finite-dimensional, then V is semisimple; this is Weyl's complete reducibility theorem. [4]
Determining whether a given subspace W is invariant under T is ostensibly a problem of geometric nature. Matrix representation allows one to phrase this problem algebraically. Write V as the direct sum W ⊕ W′; a suitable W′ can always be chosen by extending a basis of W. The associated projection operator P onto W has matrix representation
A given direct sum decomposition of into complementary subspaces still specifies a projection, and vice versa. If X {\displaystyle X} is the direct sum X = U ⊕ V {\displaystyle X=U\oplus V} , then the operator defined by P ( u + v ) = u {\displaystyle P(u+v)=u} is still a projection with range U {\displaystyle U} and kernel V {\displaystyle V} .
For the definition of the direct sum of representations please refer to the section on direct sums of representations. A representation is called isotypic if it is a direct sum of pairwise isomorphic irreducible representations. Let (,) be a given representation of a group .
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
The vector space of complex-valued class functions of a group has a natural -invariant inner product structure, described in the article Schur orthogonality relations.Maschke's theorem was originally proved for the case of representations over by constructing as the orthogonal complement of under this inner product.