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of a dense subset in the general linear group. It can be considered as a special case of the
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subgroups generalises the way a square real matrix can be written as a product of an
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of an element in algebraic group as a product of semisimple and unipotent elements
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of a Lie group as the product of semisimple, abelian, and nilpotent subgroups.
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analogues, making it harder to summarise the facts into a unified theory.
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writes a semisimple real Lie algebra as the sum of eigenspaces of a
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and associated objects, by showing how they are built up out of
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The same ideas are often applied to Lie groups, Lie algebras,
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can be regarded as a generalization of the principle of
294:writes a finite dimensional Lie algebra as a
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142:. They are essential technical tools in the
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150:; they can also be used to study the
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267:Gram–Schmidt orthogonalization
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154:of such groups and associated
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198:semisimple algebraic group
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210:Gauss–Jordan elimination
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263:upper triangular matrix
322:Birkhoff decomposition
245:of a semisimple group
173:List of decompositions
236:Iwasawa decomposition
144:representation theory
326:Bruhat decomposition
315:Bruhat decomposition
225:Cartan decomposition
187:Bruhat decomposition
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328:for affine groups.
296:semidirect product
292:Levi decomposition
265:(a consequence of
249:as the product of
156:homogeneous spaces
152:algebraic topology
146:of Lie groups and
259:orthogonal matrix
229:Cartan involution
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341:Categories
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304:semisimple
218:Weyl group
136:Lie groups
74:newspapers
255:nilpotent
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261:and an
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