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 .
