This tag is for questions relating to "Invariant Subspaces". Mathematically, an invariant subspace of a linear mapping $~T : V → V~$ from some vector space $~ V~$ to itself is a subspace $~W~$ of $~V~$ that is preserved by $~T~$.
Definition: Suppose that $~T:V\to V~$ is a linear transformation and $~W~$ is a subspace of $~V~$. Suppose further that $~T(w)\in W~$ for every $~w\in W~$. Then $~W~$ is an invariant subspace of $~V~$ relative to $~T~$.
- The subspaces $\operatorname{null}(T)$ and $\operatorname{range}(T)$ are invariant subspaces under $T$.
- $\{0\}$ and $V$ are trivial invariant subspaces.
- We do not have any special notation for an invariant subspace, so it is important to recognize that an invariant subspace is always relative to both a superspace $(V)$ and a linear transformation $(T)$, which will sometimes not be mentioned, yet will be clear from the context.
- The linear transformation involved must have an equal domain and codomain — the definition would not make much sense if our outputs were not of the same type as our inputs.
References:
https://en.wikipedia.org/wiki/Invariant_subspace
https://math.okstate.edu/people/binegar/4063-5023/4063-5023-l18.pdf
