TITLE: The Determinant of an Operator Equals the Product of Determinants of its Restrictions
QUESTION [10 upvotes]: Consider the following definitions and proved theorems

Definitions
$1$. The field $\Bbb{F}$ is $\Bbb{R}$ or $\Bbb{C}$.
$2$. $V$ is a vector space over $\Bbb{F}$.
$3$. $\mathcal{L}(V)$ is the vector space of all operators on $V$ (that is, linear maps $V \to V$) and $T \in \mathcal{L}(V)$.  
$4$. $T|_{U}$ is the restriction of $T$ to the invariant subspace $U$.
$5$. $G(\lambda_i,T)$ is the generalized eigen-space of $T$ corresponding to eigen-value $\lambda_i$. The multiplicity of $\lambda_i$ is defined to be $d_i=\dim G(\lambda_i,T)$. 
$6$. If $\Bbb{F}=\Bbb{C}$ then $\lambda_1,\cdots,\lambda_m$ are distinct eigen-values of $T$.  If $\Bbb{F}=\Bbb{R}$ then $\lambda_1,\cdots,\lambda_m$ are distinct eigen-values of  the complexification of $T$, denoted by $T_{\Bbb{C}}$. Then $\det T = \lambda_1^{d_1}\cdots\lambda_m^{d_m}$ where each $d_i$ is the multiplicity of $\lambda_i$.
Proved Theorems
$1$. Suppose $V$ is a vector space over $\Bbb{C}$ and $T \in \mathcal{L}(V)$. Let $\lambda_1,\cdots,\lambda_m$ be distinct eigen values of $T$. Then
  $V=G(\lambda_1,T) \oplus \cdots \oplus G(\lambda_m,T)$ where each $G(\lambda_i,T)$ is invariant under $T$ . Furthermore, $\dim V = d_1+\cdots+d_m$ where $d_i$ is the multiplicity of $\lambda_i$.

Now, I want to prove the following theorem just using the above tools. Any hint or help is appreciated.

Question
If $T \in \mathcal{L}(V)$ and $V=V_1 \oplus \cdots \oplus V_M$ with each $V_j,\,j=1,\cdots,M$ invariant under $T$ then $\det T = \det T|_{V_1} \cdots \det T|_{V_M}$. 

This problem happened in example $10.28$ of the book Linear Algebra Done Right. There was just one sentence for the proof of this in the text.

Because the dimensions of generalized eigen-spaces in $V_j$ add up to $\dim V$.

but I don't understand it!

REPLY [5 votes]: Motivated by the answer of Fei Li, I 
think the following is the main key point to prove this theorem.
$$\begin{align}
V &= {\Large\oplus}_{i=1}^{m}G(\lambda_i,T)
={\Large\oplus}_{i=1}^{m}\left[{\Large\oplus}_{j=1}^{M}G(\lambda_i,T|_{V_j})\right]
={\Large\oplus}_{j=1}^{M}\left[{\Large\oplus}_{i=1}^{m}G(\lambda_i,T|_{V_j})\right] \\
&={\Large\oplus}_{j=1}^{M}V_j
\end{align}
\tag{1}$$
where we have used the commutativity and associativity of direct sums and the following important identity
$$G(\lambda_i,T)={\Large\oplus}_{j=1}^{M}G(\lambda_i,T|_{V_j}) \tag{2}$$
In this way, we don't need too much complicated notation too. I don't give a complete proof of $(2)$ but I will mention the things one needs to prove it. Theorems $1$ through $4$ are mentioned to conclude theorem $5$.

Theorem 1.  If $T \in \mathcal{L}(V)$ and $\lambda \in \Bbb{F}$ and $V_j$ is a subspace of $V$ invariant under $T$ then
  $$G(\lambda,T|_{V_j}) = \{v_j \in V_j : \exists k, \, (T|_{V_j}-\lambda I|_{V_j})^{k}v_j=0\}$$
Theorem 2. If $T \in \mathcal{L}(V)$ and $\lambda \in \Bbb{F}$ and $V_j$ is a subspace of $V$ invariant under $T$ then $V_j$ is also invariant under $\lambda T$ and
  $$\lambda (T |_{V_j})=(\lambda T) |_{V_j}$$
Theorem 3. If $T_p \in \mathcal{L}(V), \, p=1,2,\cdots,n$ and $V_j$ is a subspace of $V$ invariant under each $T_p, \, p=1,2,\cdots,n$ then $V_j$ is also invariant under $\sum_{p=1}^{n}T_p$ and
  $$\sum_{p=1}^{n}T_p|_{V_j}=(\sum_{p=1}^{n}T_p)|_{V_j}$$
Theorem 4. If $T_p \in \mathcal{L}(V), \, p=1,2,\cdots,n$ and $V_j$ is a subspace of $V$ invariant under each $T_p, \, p=1,2,\cdots,n$ then $V_j$ is also invariant under ${\LARGE\circ}_{p=1}^{n}T_p$ and
  $${\LARGE\circ}_{p=1}^{n}T_p|_{V_j}=({\LARGE\circ}_{p=1}^{n}T_p)|_{V_j}$$
Theorem 5.  If $T \in \mathcal{L}(V)$ and $\lambda \in \Bbb{F}$ and $V_j$ is a subspace of $V$ invariant under $T$ then $V_j$ is also invariant under $(T-\lambda I)^{k}$ and
  $$G(\lambda,T|_{V_j}) = \{v_j \in V_j : \exists k, \, (T-\lambda I)^{k}|_{V_j}v_j=0\}$$ 
Theorem 6. If $T \in \mathcal{L}(V)$ and $\lambda \in \Bbb{F}$ and $V={\Large\oplus}_{j=1}^{M}V_j$ with each $V_j,\,j=1,\cdots,M$ invariant under $T$ then $\sum_{j=1}^{M}G(\lambda,T|_{V_j})$ is a direct sum.

Now, knowing the previous theorems, it is easy to prove what we wanted

Theorem 7. If $T \in \mathcal{L}(V)$ and $\lambda \in \Bbb{F}$ and $V={\Large\oplus}_{j=1}^{M}V_j$ with each $V_j,\,j=1,\cdots,M$ invariant under $T$ then 
  $$G(\lambda,T)={\Large\oplus}_{j=1}^{M}G(\lambda,T|_{V_j})$$