Darboux transformation of generalized Jacobi matrices

Abstract

Let J be a monic generalized Jacobi matrix, i.e. a three-diagonal block matrix of special form, introduced by M. Derevyagin and V. Derkach in 2004. We find conditions for a monic generalized Jacobi matrix J to admit a factorization J = LU with L and U being lower and upper triangular two-diagonal block matrices of special form. In this case the Darboux transformation of J defined by J (p) = UL is shown to be also a monic generalized Jacobi matrix. Analogues of Christoffel formulas for polynomials of the first and the second kind, corresponding to the Darboux transformation J (p) are found.