Invariant Subspace - Fundamental Theorem of Noncommutative Algebra

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 nontrivial invariant subspace, the fundamental theorem of noncommutative algebra asserts that Lat(Σ) contains nontrivial elements for certain Σ.

Theorem (Burnside) Assume V is a complex vector space of finite dimension. For every proper subalgebra Σ of L(V), Lat(Σ) contain a nontrivial element.

Burnside's theorem is of fundamental importance in linear algebra. One consequence is that every commuting family in L(V) can be simultaneously upper-triangularized.

A nonempty Σ ⊂ L(V) is said to be triangularizable if there exists a basis {e₁...e_n} of V such that

In other words, Σ is triangularizable if there exists a basis such that every element of Σ has an upper-triangular matrix representation in that basis. It follows from Burnside's theorem that every commutative algebra Σ in L(V) is triangularizable. Hence every commuting family in L(V) can be simultaneously upper-triangularized.