Direct sums of invariant subspaces Let $A$ be a complex $n\times n$ matrix, with its Jordan carnonical form as $J=diag(J_1,\cdots,J_s)$. Then there exists an invertible matrix $P$ such that $P^{-1}AP=J$. It is easy to verify that $\Bbb C^n=V_1\oplus \cdots \oplus V_s$ with $V_i$ is the columns of $P$ correspoing to $J_i$. 
My question is that: for any invariant subspace $V\subset \Bbb C^n$, can we have $V=(V\cap V_1)\oplus\cdots\oplus (V\cap V_s)?$$
If all $J_i$ is of $1\times 1$ matrix, then it is easy. But for general case, I do not have an idea.

This is false. Let $A$ be the $2\times 2$ identity matrix. Then $V_1 = span{\begin{pmatrix} 1 \\ 0 \end{pmatrix}}$, and $V_2 = span{\begin{pmatrix} 0 \\ 1 \end{pmatrix}}$ are the invariant subspaces corresponding to the Jordan blocks of $A$, and $\mathbb{C}^2 = V_1 \oplus V_2$.
Let $V = span{\begin{pmatrix} 1 \\ 1 \end{pmatrix}}$.
Then $V \cap V_1 = 0 = V \cap V_2$.
So $V \neq (V\cap V_1) \oplus (V \cap V_2)$.
