TITLE: Formula relating traces of a linear map, its restriction to an invariant subspace and the induced quotient map?
QUESTION [6 upvotes]: Let $V$ be a finite-dimensional vector space, $T\colon V \to V$ a linear map and $W \subset V$ a $T$-invariant subspace (i.e., $T(W) \subset W$). Then there is a well-defined induced quotient map $\overline{T}\colon V/W \to V/W$.
Now I recall having seen the following determinant formula: $\det(T) = \det(\overline{T}) \det(T|_W)$, where $T|_W\colon W \to W$ is the restriction of $T$ to the subspace $W$ (though I do not remember the proof anymore).

Is there some similar formula relating the traces of $T$, $\overline{T}$ and $T|_W$?

REPLY [6 votes]: The idea is the same. Choose a basis $\mathfrak{B}$ of $W$, and extend it to a basis $\mathfrak{B} \cup \mathfrak{C}$ of $V$, where $\mathfrak{C} \subseteq V \setminus W$. With respect to this basis, the matrix of $T$ will be of the block form
$$
\begin{bmatrix}
B & A\\
0 & C
\end{bmatrix},
$$
where $A, B, C$ are suitably-sized matrices, with $B$ and $C$ square matrices.
Now $B$ represents the action of $T$ on $W$ with respect to $\mathfrak{B}$, and $C$ the action of $T$ on $V/W$, with respect to the basis given by the image of $\mathfrak{C}$. Hence the determinant formula, and the analogue
$$
\operatorname{tr}(T) = \operatorname{tr}(T|_W) + \operatorname{tr}(\overline{T}). 
$$
for the trace.