A possible relation between subspaces \mathcal{M} of a vector space and linear transformations A on that space is invariance. We say that \mathcal{M} is invariant under A , if x in \mathcal{M} implies that A x is in \mathcal{M} . (Observe that the implication relation is required in one direction only; we do not assume that every y in \mathcal{M} can be written in the form y=A x with x in \mathcal{M} ; we do not even assume that A x in \mathcal{M} implies x in \mathcal{M} . Presently we shall see examples in which the conditions we did not assume definitely fail to hold.) We know that a subspace of a vector space is itself a vector space; if we know that \mathcal{M} is invariant under A , we may ignore the fact that A is defined outside \mathcal{M} and we may consider A as a linear transformation defined on the vector space \mathcal{M} . Invariance is often considered for sets of linear transformations, as well as for a single one; \mathcal{M} is invariant under a set if it is invariant under each member of the set.
What can be said about the matrix of a linear transformation A on an n -dimensional vector space \mathcal{V} if we know that some \mathcal{M} is invariant under A ? In other words: is there a clever way of selecting a basis \mathcal{X}=\{x_{1}, \ldots, x_{n}\} in \mathcal{V} so that [A]=[A; \mathcal{X}] will have some particularly simple form? The answer is in Section: Dimension of a subspace , Theorem 2; we may choose \mathcal{X} so that x_{1}, \ldots, x_{m} are in \mathcal{M} and x_{m+1}, \ldots, x_{n} are not. Let us express A x_{j} in terms of x_{1}, \ldots, x_{n} . For m+1 \leq j \leq n , there is not much we can say: A x_{j}=\sum_{i} \alpha_{i j} x_{i} . For 1 \leq j \leq m , however, x_{j} is in \mathcal{M} , and therefore (since \mathcal{M} is invariant under A ) A x_{j} is in \mathcal{M} . Consequently, in this case A x_{j} is a linear combination of x_{1}, \ldots, x_{m} ; the \alpha_{i j} with m+1 \leq i \leq n are zero. Hence the matrix [A] of A , in this coordinate system, will have the form [A]=\begin{bmatrix} {[A_{1}]} & {[B_{0}]} \\ {[0]} & {[A_{2}]} \end{bmatrix}, where [A_{1}] is the ( m -rowed) matrix of A considered as a linear transformation on the space \mathcal{M} (with respect to the coordinate system \{x_{1}, \ldots, x_{m}\} ), [A_{2}] and [B_{0}] are some arrays of scalars (in size (n-m) by (n-m) and m by (n-m) respectively), and [0] denotes the rectangular ( (n-m) by m ) array consisting of zeros only. (It is important to observe the unpleasant fact that [B_{0}] need not be zero.)
