The structure of the eigenvectors of sparse matrices

Abstract

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 λ 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 λ 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.