91Ó°ÊÓ

Let \(T\) be a linear operator on a finite-dimensional vector space \(V\), and let \(\mathrm{W}_{1}, \mathrm{~W}_{2}, \ldots, \mathrm{W}_{k}\) be \(\mathrm{T}\)-invariant subspaces of \(\mathrm{V}\) such that \(\mathrm{V}=\mathrm{W}_{1} \oplus \mathrm{W}_{2} \oplus \cdots \oplus \mathrm{W}_{k}\). Prove that $$ \operatorname{det}(T)=\operatorname{det}\left(T_{W_{1}}\right) \cdot \operatorname{det}\left(T_{W_{2}}\right) \cdots \cdots \operatorname{det}\left(T_{W_{k}}\right) . $$

Let \(\mathcal{C}\) be a collection of diagonalizable linear operators on a finitedimensional vector space V. Prove that there is an ordered basis \(\beta\) such that \([T]_{\beta}\) is a diagonal matrix for all \(T \in \mathcal{C}\) if and only if the operators of \(\mathcal{C}\) commute under composition. (This is an extension of Exercise 25.) Hints for the case that the operators commute: The result is trivial if each operator has only one eigenvalue. Otherwise, establish the general result by mathematical induction on \(\operatorname{dim}(\mathrm{V})\), using the fact that \(V\) is the direct sum of the eigenspaces of some operator in \(\mathcal{C}\) that has more than one eigenvalue.

Let \(B_{1}, B_{2}, \ldots, B_{k}\) be square matrices with entries in the same field, and let \(A=B_{1} \oplus B_{2} \oplus \cdots \oplus B_{k}\). Prove that the characteristic polynomial of \(A\) is the product of the characteristic polynomials of the \(B_{i}\) 's.