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We study the eigenvalue problem for some special class of anti-triangularmatrices. Though the eigenvalue problem is quite classical, as far as we know,almost nothing is known about properties of eigenvalues for anti-triangularmatrices. In this paper, we show that there is a nice class of anti-triangularmatrices whose eigenvalues are given explicitly by their elements. Moreover,this class contains several interesting subclasses which we characterize interms of probability measures. We also discuss the application of our maintheorem to the study of interacting particle systems, which are stochasticprocesses studied in extensive literature.

Eigenvalue problem for some special class of anti-triangular matrices