Let $V$ be an $n$-dimensional vector space over $\mathbb{F}$ and $T : V → V$ be linear. We say that $T$ stabilizes a maximal flag $\{V_i\}_{i=0}^n$ in $V$ if $T (V_i ) ⊆ V_i$ , for all $i = 1, . . . , n$. Show that, $T$ stabilizes a maximal flag in $V$ if and only if $V$ can be expressed as the direct sum of its generalised eigenspaces.
I have proved the "if" direction, and I have some trouble proving the "iff" direction.
My work: Let $\{v_1,v_2,...,v_n\}$ be a basis of $V_n=V$ such that $\{v_1,v_2,...,v_i\}$ is a basis of $V_i$, where $\{0\} = V_0 < V_1 < V_2...<V_n=V$ is a flag of subspaces on $V$. The matrix of $T$ with respect to this basis is upper triangular, so that $\det(xI-T) = \Pi_{i=1}^n (x-a_{ii})$. Since the characteristic polynomial of $T$ splits over $\mathbb{F}$, so does the minimal polynomial of $T$ (the minimal polynomial divides the characteristic polynomial). As a result, we have $V = \bigoplus_{i=1}^k V(\lambda_i)$.
Now for the other direction, I begin with $V = \bigoplus_{i=1}^k V(\lambda_i)$ as the direct sum decomposition of $V$ over the generalized eigenspaces. So, the minimal polynomial of $T$ splits over $\mathbb{F}$. What do I do next? I know that the matrix of $T$ can be brought to its Jordan form. I am wondering if a basis consisting of the columns of the Jordan form of $T$ would suffice? I am not sure!
I need to create a chain of subspaces $\{0\} = V_0 < V_1 < V_2...<V_n=V$ which are $T$-invariant, to solve the problem. How do I arrive at one?
Edit and Update: Since $V$ can be written as the sum of generalized eigenspaces of $T$, the minimal polynomial of $T$ splits over $\mathbb{F}$. So there is a Jordan basis for $T$, w.r.t which the matrix of $T$ is in the Jordan form (obviously). Now, we construct $V_i$ in the following manner, where $J[1],J[2],..,J[n]$ denote the columns of $J$.
$$V_i = \text{span}(J[1],J[2],...,J[i])$$
Does this work? I think that the $V_i$'s are properly contained in $V_{i+1}$, since the columns of $J$ are linearly independent (right?). As a result, we obtain a chain of size $n$. Also it seems like all these $V_i$ are $T$-invariant because the Jordan form $J$ is upper triangular.