Linear algebra MOC

Invariant subspace

An invariant subspace of a Linear endomorphism 𝑇 :𝑉 →𝑉 is a vector subspace 𝑊 ⊆𝑉 that is preserved by 𝑇, i.e. 𝑇𝑤 ∈𝑊 for all 𝑤 ∈𝑊. linalg It follows every eigenspace is also an invariant subspace. Every linear endomorphism has two trivial invariant subspaces, namely the null space {⃗𝟎} and the full space 𝑉. Any other invariant subspace is nontrivial.

Jordan canonical form essentially decomposes a matrix into operators on invariant subspaces. In general, if 𝑉 =𝑊 ⊕𝑊′ we can reduce 𝑇 to 𝑈 ⊕𝑈′, where 𝑈 :𝑊 →𝑊 and 𝑈′ :𝑊′ →𝑊′.

Representations

For collections of linear endomorphisms, such as a Group representation, an invariant subspace is preserved by all members of the collection. Let Γ :𝐺 →GL(𝑉) be a representation and 𝑊 ⊆𝑉 be a subspace. Then 𝑊 is Γ-invariant iff Γ(𝑔)𝑤 ∈𝑊 for all 𝑔 ∈𝐺 and 𝑤 ∈𝑊. rep

A representation with no non-trivial invariant subspaces is called irreducible.

Properties

• The orthogonal complement of an invariant subspace under a unitary operator is invariant

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