On a matrix with integer eigenvalues and its relation to conditional Poisson sampling

Abstract

A special non-symmetric N × N matrix with eigenvalues 0, 1, 2, . . . ,N − 1 is presented. The matrix appears in sampling theory. Its right eigenvectors, if properly normalized, give the inclusion probabilities of the Conditional Poisson design (for all different fixed sample sizes). The explicit expressions for the right eigenvectors become complicated for N large. Nevertheless, the left eigenvectors have a simple analytic form. An inversion of the left eigenvector matrix produces the right eigenvectors − the inclusion probabilities. Finally, a more general matrix with similar properties is defined and expressions for its left and right eigenvectors are derived.