Reducibility

A particularly important subcase of the notion of invariance is that of reducibility. If and are two subspaces such that both are invariant under and such that is their direct sum, then is reduced (decomposed) by the pair . The difference between invariance and reducibility is that, in the former case, among the collection of all subspaces invariant under we may not be able to pick out any two, other than and , with the property that is their direct sum. Or, saying it the other way, if is invariant under , there are, to be sure, many ways of finding an such that , but it may happen that no such will be invariant under .

The process described above may also be turned around. Let and be any two vector spaces, and let and be any two linear transformations (on and respectively). Let be the direct sum ; we may define on a linear transformation called the direct sum of and , by writing We shall omit the detailed discussion of direct sums of transformations; we shall merely mention the results. Their proof are easy. If reduces , and if we denote by the linear transformation considered on alone, and by the linear transformation considered on alone, then is the direct sum of and . By suitable choice of basis (namely, by choosing in and in ) we may put the matrix of the direct sum of and in the form displayed in the preceding section, with , , and . If is any polynomial, and if we write , , then the direct sum of and will be .

EXERCISES