We discuss the following question: Given the zero-nonzero pattern of a matrix A with complex entries, what can be said about the zero-nonzero pattern of its eigenvectors? To be more general, what are the possible sparsity patterns for the bases of the maximal invariant subspaces of A associated with each of its eigenvalues? Or even, what is the sparsity pattern of the similarity transformation M such that M-1 AM is in Jordan canonical form, i.e., A's Jordan basis? Let struct(A) be the usual directed graph associated with the zero-nonzero pattern of A. The main result of this paper is that there exists a matrix B such that if lambda is an eigenvalue of A with algebraic multiplicity m, then there are m columns of B that form a basis for the maximal invariant subspace of A associated with lambda and such that struct(B) is a subgraph of the graph obtained by adding all the edges of the form (i, i) to the transitive closure of struct(A), which we call rstruct(A). We show that if the defective eigenvalues of A have geometric multiplicity one, then the matrix B above can be chosen in such a way that there exists a permutation matrix P for which BP is a Jordan basis of A. We present examples and theorems showing that our results are sharp. Similar results hold for the real Jordan canonical form.207120