ABSTRACT
In this chapter, we deal only with square matrices. Suppose we have a matrix A in Cn×n and a subspace M ⊆ Cn×n. Recall, M is an invariant subspace for A if A(M) ⊆ M. That is, when A multiplies a vector v in M , the result Av remains in M . It is easy to see, for example, that the null space of A is an invariant subspace for A. For the moment, we restrict to one-dimensional subspaces. Suppose v is a nonzero vector inCn and sp(v) is the one-dimensional subspace spanned by v. Then, to have an invariant subspace of A, we would need to have A(sp(v)) ⊆ sp(v). But v is in sp(v), so Av must be in sp(v) also. Therefore, Av would have to be a scalar multiple of v. Let’s say Av = v. Conversely, if we can find a nonzero vector v and a scalar with Av = v, then sp(v) is an invariant subspace for A. So the search for one-dimensional invariant subspaces for a matrix A boils down to solving Av = v for a nonzero vector v. This leads to some language.
