(Massey University, 2005) Bondesson, L.; Traat, I.
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.
