Reducibility

A particularly important subcase of the notion of invariance is that of reducibility. If $ℳ$ and $𝒩$ are two subspaces such that both are invariant under $A$ and such that $𝒱$ is their direct sum, then $A$ is reduced (decomposed) by the pair $(ℳ,𝒩)$ . 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 $𝒪$ and $𝒱$ , with the property that $𝒱$ is their direct sum. Or, saying it the other way, if $ℳ$ is invariant under $A$ , there are, to be sure, many ways of finding an $𝒩$ such that $𝒱=ℳ\oplus 𝒩$ , but it may happen that no such $𝒩$ will be invariant under $A$ .

The process described above may also be turned around. Let $ℳ$ and $𝒩$ be any two vector spaces, and let $A$ and $B$ be any two linear transformations (on $ℳ$ and $𝒩$ respectively). Let $𝒱$ be the direct sum $ℳ\oplus 𝒩$ ; we may define on $𝒱$ a linear transformation $C$ called the direct sum of $A$ and $B$ , by writing $$Cz=C(x,y)=(Ax,By).$$ We shall omit the detailed discussion of direct sums of transformations; we shall merely mention the results. Their proof are easy. If $(ℳ,𝒩)$ reduces $C$ , and if we denote by $A$ the linear transformation $C$ considered on $ℳ$ alone, and by $B$ the linear transformation $C$ considered on $𝒩$ alone, then $C$ is the direct sum of $A$ and $B$ . By suitable choice of basis (namely, by choosing $x_1,\dots ,x_m$ in $ℳ$ and $x_{m+1},\dots ,x_n$ in $𝒩$ ) 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