Invariance

A possible relation between subspaces of a vector space and linear transformations on that space is invariance. We say that is invariant under , if in implies that is in . (Observe that the implication relation is required in one direction only; we do not assume that every in can be written in the form with in ; we do not even assume that in implies in . 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 is invariant under , we may ignore the fact that is defined outside and we may consider as a linear transformation defined on the vector space . Invariance is often considered for sets of linear transformations, as well as for a single one; 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 on an -dimensional vector space if we know that some is invariant under ? In other words: is there a clever way of selecting a basis in so that will have some particularly simple form? The answer is in Section: Dimension of a subspace , Theorem 2; we may choose so that are in and are not. Let us express in terms of . For , there is not much we can say: . For , however, is in , and therefore (since is invariant under ) is in . Consequently, in this case is a linear combination of ; the with are zero. Hence the matrix of , in this coordinate system, will have the form where is the ( -rowed) matrix of considered as a linear transformation on the space (with respect to the coordinate system ), and are some arrays of scalars (in size by and by respectively), and denotes the rectangular ( by ) array consisting of zeros only. (It is important to observe the unpleasant fact that need not be zero.)