A particularly important subcase of the notion of invariance is that of reducibility. If \(\mathcal{M}\) and \(\mathcal{N}\) are two subspaces such that both are invariant under \(A\) and such that \(\mathcal{V}\) is their direct sum, then \(A\) is reduced (decomposed) by the pair \((\mathcal{M}, \mathcal{N})\) . The difference between invariance and reducibility is that, in the former case, among the collection of all subspaces invariant under \(A\) we may not be able to pick out any two, other than \(\mathcal{O}\) and \(\mathcal{V}\) , with the property that \(\mathcal{V}\) is their direct sum. Or, saying it the other way, if \(\mathcal{M}\) is invariant under \(A\) , there are, to be sure, many ways of finding an \(\mathcal{N}\) such that \(\mathcal{V}=\mathcal{M} \oplus \mathcal{N}\) , but it may happen that no such \(\mathcal{N}\) will be invariant under \(A\) .
The process described above may also be turned around. Let \(\mathcal{M}\) and \(\mathcal{N}\) be any two vector spaces, and let \(A\) and \(B\) be any two linear transformations (on \(\mathcal{M}\) and \(\mathcal{N}\) respectively). Let \(\mathcal{V}\) be the direct sum \(\mathcal{M} \oplus \mathcal{N}\) ; we may define on \(\mathcal{V}\) a linear transformation \(C\) called the direct sum of \(A\) and \(B\) , by writing \[C z=C(x, y)=(A x, B y).\] We shall omit the detailed discussion of direct sums of transformations; we shall merely mention the results. Their proof are easy. If \((\mathcal{M}, \mathcal{N})\) reduces \(C\) , and if we denote by \(A\) the linear transformation \(C\) considered on \(\mathcal{M}\) alone, and by \(B\) the linear transformation \(C\) considered on \(\mathcal{N}\) alone, then \(C\) is the direct sum of \(A\) and \(B\) . By suitable choice of basis (namely, by choosing \(x_{1}, \ldots, x_{m}\) in \(\mathcal{M}\) and \(x_{m+1}, \ldots, x_{n}\) in \(\mathcal{N}\) ) we may put the matrix of the direct sum of \(A\) and \(B\) in the form displayed in the preceding section, with \([A_{1}]=[A]\) , \([B_{0}]=[0]\) , and \([A_{2}]=[B]\) . If \(p\) is any polynomial, and if we write \(A^{\prime}=p(A)\) , \(B^{\prime}=p(B)\) , then the direct sum \(C^{\prime}\) of \(A^{\prime}\) and \(B^{\prime}\) will be \(p(C)\) .
EXERCISES
Exercise 1. Give an example of a linear transformation \(A\) on a finite-dimensional vector space \(\mathcal{V}\) such that \(\mathcal{O}\) and \(\mathcal{V}\) are the only subspaces invariant under \(A\) .
Exercise 2. Let \(D\) be the differentiation operator on \(\mathcal{P}_n\) . If \(m \leq n\) , then the subspace \(\mathcal{P}_m\) is invariant under \(D\) . Is \(D\) on \(\mathcal{P}_{m}\) invertible? Is there a complement of \(\mathcal{P}_m\) in \(\mathcal{P}_n\) such that it together with \(\mathcal{P}_{m}\) reduces \(D\) ?
Exercise 3. Prove that the subspace spanned by two subspaces, each of which is invariant under some linear transformation \(A\) , is itself invariant under \(A\) .