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This paper is concerned with spectral problems of second-order vector difference equations with two-point boundary value conditions, where the matrix-valued coefficient of the leading term may be singular. A concept of self-adjointness of the boundary value conditions is introduced. The self-adjointness of the corresponding difference operator is discussed on a suitable admissible function space, and fundamental spectral results are obtained. The dual orthogonality of eigenfunctions is shown in a special case. Rayleigh's principles and the minimax theorems in two linear spaces are given. As an application, a comparison theorem for eigenvalues of two Sturm᎐Liouville problems is presented. ᮊ 1999 Academic Press

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