Editing Invariant subspace (section)

== For multiple operators ==
Given a collection {{Math|{{mathcal|T}}}} of operators, a subspace is called {{Math|{{mathcal|T}}}}-invariant if it is invariant under each {{Math|''T'' &isin; {{mathcal|T}}}}.  

As in the single-operator case, the invariant-subspace lattice of {{Math|{{mathcal|T}}}}, written {{Math|Lat({{mathcal|T}})}}, is the set of all {{Math|{{mathcal|T}}}}-invariant subspaces, and bears the same meet and join operations.  Set-theoretically, it is the intersection <math display="block">\mathrm{Lat}(\mathcal{T})=\bigcap_{T\in\mathcal{T}}{\mathrm{Lat}(T)}\text{.}</math>

=== Examples ===
Let {{Math|End(''V'')}} be the set of all linear operators on {{Mvar|V}}.  Then {{Math|1=Lat(End(''V''))={0,''V''}<nowiki />}}.  

Given a [[Group representation|representation]] of a [[group (mathematics)|group]] ''G'' on a vector space ''V'', we have a linear transformation ''T''(''g'') : ''V'' → ''V'' for every element ''g'' of ''G''. If a subspace ''W'' of ''V'' is invariant with respect to all these transformations, then it is a [[subrepresentation]] and the group ''G'' acts on ''W'' in a natural way.  The same construction applies to [[Algebra representation|representations of an algebra]].  

As another example, let {{Math|''T'' &isin; End(''V'')}} and {{Mvar|&Sigma;}} be the algebra generated by {1,&thinsp;''T''&thinsp;}, where 1 is the identity operator. Then Lat(''T'') = Lat(Σ). 

=== Fundamental theorem of noncommutative algebra ===
Just as the fundamental theorem of algebra ensures that every linear transformation acting on a finite-dimensional complex vector space has a non-trivial invariant subspace, the ''fundamental theorem of noncommutative algebra'' asserts that Lat(Σ) contains non-trivial elements for certain Σ.
{{Math theorem
| math_statement = Assume {{mvar|V}} is a complex vector space of finite dimension. For every proper subalgebra {{mvar|Σ}} of {{math|End(''V'')}}, {{math|Lat(''Σ'')}} contains a non-trivial element.
| note = Burnside
}}
One consequence is that every commuting family in ''L''(''V'') can be simultaneously [[upper-triangular|upper-triangularized]].  To see this, note that an upper-triangular matrix representation corresponds to a [[Flag (linear algebra)|flag]] of invariant subspaces, that a commuting family generates a commuting algebra, and that {{Math|End(''V'')}} is not commutative when {{Math|dim(''V'') &geq; 2}}.