Linear algebra MOC

Invariant subspace

An invariant subspace of a Linear endomorphism is a vector subspace that is preserved by , i.e. for all . #m/def/linalg It follows every eigenspace is also an invariant subspace. Every linear endomorphism has two trivial invariant subspaces, namely the null space and the full space . Any other invariant subspace is nontrivial.

Jordan canonical form essentially decomposes a matrix into operators on invariant subspaces. In general, if we can reduce to , where and .

Representations

For collections of linear endomorphisms, such as a Group representation, an invariant subspace is preserved by all members of the collection. Let be a representation and be a subspace. Then is -invariant iff for all and . #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

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