Invariant subspace

An invariant subspace of a Linear endomorphism $𝑇 : 𝑉 \to 𝑉$ is a vector subspace $𝑊 \subseteq 𝑉$ that is preserved by $𝑇$ , i.e. $𝑇 𝑤 \in 𝑊$ for all $𝑤 \in 𝑊$ . #m/def/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 $𝑉 = 𝑊 \oplus 𝑊'$ we can reduce $𝑇$ to $𝑈 \oplus 𝑈'$ , where $𝑈 : 𝑊 \to 𝑊$ and $𝑈' : 𝑊' \to 𝑊'$ .

Representations

For collections of linear endomorphisms, such as a Group representation, an invariant subspace is preserved by all members of the collection. Let $Γ : 𝐺 \to G L (𝑉)$ be a representation and $𝑊 \subseteq 𝑉$ be a subspace. Then $𝑊$ is $Γ$ -invariant iff $Γ (𝑔) 𝑤 \in 𝑊$ for all $𝑔 \in 𝐺$ and $𝑤 \in 𝑊$ . #m/def/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

#state/tidy | #lang/en | #SemBr