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Let J be a monic Jacobi matrix associated with the Cauchy transform F of aprobability measure. We construct a pair of the lower and upper triangularblock matrices L and U such that J=LU and the matrix J_c=UL is a monicgeneralized Jacobi matrix associated with the function F_c(z)=zF(z)+1. It turnsout that the Christoffel transformation J_c of a bounded monic Jacobi matrix Jcan be unbounded. This phenomenon is shown to be related to the effect ofaccumulating at infinity of the poles of the Pad\'e approximants of thefunction F_c although F_c is holomorphic at infinity. The case of theUL-factorization of J is considered as well.

Darboux transformations of Jacobi matrices and Pad\'e approximation