An invariant subspace of a Linear endomorphism is a vector subspace that is preserved by , i.e. for all . 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 . rep
A representation with no non-trivial invariant subspaces is called irreducible.