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...Invariant (mathematics)|invariant]] each [[Linear subspace|subspace]] left invariant by every operator in ''A''. but has the same invariant subspaces, so it is not reflexive. ...

...plete lattice]]. Since the [[orthogonal projection]]s corresponding to the subspaces in a nest [[commutative operation|commute]], nests are commutative subspace ...p;&times;&nbsp;''n'' complex matrices ''M'' leaving each subspace in ''N'' invariant &nbsp; that is, satisfying <math>MS\subseteq S</math> for each ''S'' in ''N ...

...|subspace]] ''W'' of ''V'' that is preserved by ''T''. More generally, an invariant subspace for a collection of linear mappings is a subspace preserved by eac ...n '''invariant subspace for''' <math>T</math>, or equivalently, {{Mvar|T}}-invariant, if {{Mvar|T}} transforms any vector <math>\mathbf{v} \in W</math> back int ...

...anslation of synthetic geometric statements into [[Invariant (mathematics)|invariant]] algebraic statements. This can create a useful framework for the modeling ...

...eflexive operator algebra]], an operator algebra that has enough invariant subspaces to characterize it ...

...cts a minimal decomposition of the [[vector space]] into [[linear subspace|subspaces]] that are cyclic for ''A'' (i.e., [[linear span|spanned]] by some vector a ...lic subspaces, the primary form decomposes into a maximum number of cyclic subspaces. It is also defined over ''F'', but has somewhat different properties: find ...

...f ''A'', then '''C'''^''n'' can be decomposed into the invariant subspaces of ''A'' ...'Ran''(''I'' − ''C'')^''n''. Consider the decreasing sequence of subspaces ...

...ace)|finite-dimensional]] [[vector space]] has an eigenvector, solving the invariant subspace problem for these spaces.]] ..., each example, found so far, of an operator with no non-trivial invariant subspaces is an operator that acts on a Banach space that is not [[isomorphic]] to a ...

It is clear that ''K''₁ and ''K''₂ are invariant subspaces of ''V''. where each ''H_α'' is an invariant subspaces of ''V'' and ''V'' restricted to each ''H_α'' is the unilateral s ...

...any [[Isometry (quadratic forms)|isometry]] between two [[Linear subspace|subspaces]] of a nonsingular [[quadratic space]] over a [[field (algebra)|field]] ''k ...orm]]. If {{nowrap|''f'' : ''U'' → ''U'''}} is an [[isometry]] between two subspaces of ''V'' then ''f'' extends to an isometry of ''V''.{{sfn|Roman|2008|p=275- ...

...</math> in a similar way; this algebra is a [[Direct sum of modules|sum of subspaces]] ''gC''[''N''] as ''g'' runs through the elements of ''G'', and is the gro <math>A \rtimes G</math>, which is the sum of subspaces ...

...moncini|first=Valeria|author-link=Valeria Simoncini|year=2015|title=Krylov Subspaces|editor=Nicholas J. Higham|display-editors=etal|encyclopedia=The Princeton C ...) = r_0</math>. So <math>r_0</math> is the maximal dimension of the Krylov subspaces <math>\mathcal{K}_r(A,b)</math>. ...

...'' induces an [[Group action (mathematics)|action]] of ''G'' on the vector subspaces ''W'' of ''H'' in this way: Let ''X'' be a set of subspaces of ''H'' such that ...

...math>. Excluding trivial cases, there are an infinite number of horizontal subspaces at each point. Also note that arbitrary choices of horizontal space at each ...''G''-bundle]], then the horizontal bundle is usually required to be ''G''-invariant: such a choice is equivalent to a [[Connection (principal bundle)|connectio ...

...(\mathbb{R})</math> consists of a [[sequence]] of nested [[linear subspace|subspaces]] ...f-similarity'' in ''time'' demands that each subspace ''V_k'' is invariant under shifts by [[integer]] [[multiple (mathematics)|multiple]]s of ''2<sup ...

...often called a Zen space<ref name="Jacob"/>) with respect to a translation-invariant positive regular [[Borel measure]] <math>\nu</math> on the closed right com ...le=A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces|publisher=Bulletin of the London Mathematical Society|date=2007-06-03|volum ...

...ogy are [[homeomorphism]]s and [[homotopy|homotopies]]. A property that is invariant under such deformations is a [[topological property]]. Basic examples of to ...

...v.pdf}}</ref> He won the 1985 Junior [[Berwick Prize]] for his work on the invariant subspace problem.<ref>{{cite journal|title=Records of Proceedings at Meetin ...

where vectors and real scalar numbers are invariant under This is due to both scalars and vectors being invariant to reversion ( it is impossible ...

...composition implies that there exists a nested sequence of ''A''-invariant subspaces {{math|1={0} = ''V''₀ ⊂ ''V''₁ ⊂ ⋯ ⊂ ''V_n' ...Notice the preimage of ''W_&mu;'' under the quotient map is an [[invariant subspace]] of ''A'' that contains ''V_&lambda;''. Continue this w ...