What is a characteristic subspace (w.r.t a linear operator)? I am working through a book on linear operators and, in the context of a subspace being invariant under an operator, they mention "characteristic subspaces".
"All characteristic subspaces of an operator are invariant"

The book never defined such a term, and a quick google search didn't really provide me with a formal definition. I assume it is related to characteristic functions in some way?
 A: Given a vector space $V$ over a field $\mathbb{K}$  and an a fixed endomorphism $f\in \mathrm{End}\left(V\right)$ we can define an homomorphism between the ring of polynomials $\mathbb{K}\left[t\right]$ and the vector space of endomorphisms $\mathrm{End}\left(V\right)$:
$$\mathbb{K}\left[t\right]\ni\varphi(t)=a_0 + a_1t+\dots a_nt^n\mapsto\phi(f)=a_0\mathrm{id}+a_1f+\dots a_nf^n$$
where $\mathrm{id}$ stands for the identity endomorphism and $f^n=\underset{n \text{ times}}{f\circ f\circ\dots\circ f}$.
The minimal polynomial associated to $f$ is a polynomial $\phi_\mathrm{m}(t)$ such that:


*

*$\phi_\mathrm{m}\left(f\right)=0$ (that is, the associated endomorphism is the zero endomorphism)

*every other polynomial $\varphi(t)$ such that $\varphi\left(f\right)=0$ is a multiple of $\phi_\mathrm{m}(t)$.


This minimal polynomial can be factored in prime polynomials in general, i.e. $1-t^2=(1+t)(1-t)$
If $p(t)$ is one of this prime factors of the minimal polynomial, then the associated characteristic subspace $V_p$ is defined as:
$$
V_p = \left\lbrace v\in\mathrm{Ker}\left(p^k\left(f\right)\right),\text{ for some $k\in \mathbb{N}$}\right\rbrace
$$
that is $$V_p = \bigcup_{k=0}^\infty\mathrm{Ker}\left(p^k\left(f\right)\right)$$
So there you have it, that is a characteristic subspace. 
Are they $f$-invariant? yes. Why? because if $v\in\mathrm{Ker}\left(\varphi\left(f\right)\right)$ then $f(v)\in\mathrm{Ker}\left(\varphi\left(f\right)\right)$. Ask if you need me to elaborate on this.
A: Sometimes, especially in the context of ODEs, the equation $\det(A - \lambda I) = 0$ is called the "characteristic equation" of $A$, the eigenvalues (which are the solutions of this equation) are called the "characteristic values" and the eigenspaces are called the "characteristic spaces". This makes sense in the context of your quote as indeed the eigenspaces are invariant subspaces.
A: In the French speaking community, the notion of "Characteristic subspace" (=sous-espace caractéristique) is taught in every advanced textbook on linear algebra. Linguistically, it corresponds to the notion of "generalised eigenspace" in the US textbooks:
https://fr.wikipedia.org/wiki/Sous-espace_caractéristique (in French)
https://en.wikipedia.org/wiki/Generalized_eigenvector
For the book you are referring too, I suspect either a poor translation, or a foreign author not fully aware of the subtleties of the mathematical English language ;-).
