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Examples of direct sum
direct sum
With the Vs, though, it's extremely likely to be the direct sum of vector spaces.
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If a representation is the direct sum of two proper nontrivial subrepresentations, it is said to be decomposable.
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Diagonalizable matrices can be decomposed into a direct sum of eigenspaces, as per the eigendecomposition of a matrix.
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The direct sum of two representations carries no more information about the group G than the two representations do individually.
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This usage faces the objection that this same symbol is already used in mathematics for the direct sum of algebraic structures.
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Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras.
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The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives.
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In the discrete case, for example, instead of finding a complete eigenbasis, it is a bit more convenient to write the Hilbert space as a direct sum of eigenspaces.
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The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum.
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