Partly zero eigenvectors

This paper is concerned with the forced presence or absence of zero components in an eigenvector. Relative to a fixed matrix Awith eigenvalue λ, we characterize the strictly nonzero part of a partly zero eigenvector associated with λ. We also give a sufficient condition for a fixed matrix to have a partly zero eigenvector, and discuss several examples in which a matrix has one or more partly zero eigenvectors. Our main results, however, are qualitative in nature. We associate a zero/nonzero pattern class of matrices with a digraph, and characterize the set of pattern classes which requires all eigenvectors to be strictly nonzero. A sufficient condition is also given that identifies the components of a partly zero eigenvector which may be nonzero.