# Linear Transform T = A + B. When is Tw = Aw + Bw such that Tw is T restricted to the domain W

I have a question related to the Jordan-Chevalley Decomposition but I am also wondering about the general case.

I have that V is a finite dimensional vector space over $$\mathbb{C}$$ and $$T:V\rightarrow V$$linear. $$T = D_{T}+N_{T}$$ with $$D_{T}$$ diagonalizable and $$N_{T}$$ nilpotent and $$D_{T}N_{T}=N_{T}D_{T}$$.

I have already proved that if $$W\subset V$$ is T-Invariant then W is also $$D_{T}$$-Invariant and $$N_{T}$$-Invariant.

However, what I am struggling to prove is that if we were to restrict the domains of these functions then $$T\mid_{w} = D_{T}\mid_{w}+N_{T}\mid_{w}$$

Finally, in general, for arbitrary $$T:V\rightarrow V$$linear with $$T = A + B$$, A&B also linear, when is it that $$T\mid_{w} = A\mid_{w}+B\mid_{w}$$ for $$W\subset V$$

## 1 Answer

You struggle witj a triviality.

If $$w\in W$$ then $$T|_W(w)=T(w)=A(w)+B(w)=A|_W(w)+B|_W(w)$$