Cauchy operator of a non-stationary linear differential equation with a small parameter at the derivative

A diagonalization algorithm for a matrix pencil depending on a variable and a parameter in the cases when the limiting matrix has a simple spectrum or a multiple eigenvalue for all values of the variable is put forward. The algorithm uses an exhaustive superposition of special similarity transformations. Formulae for the Cauchy operator of a linear non-stationary equation with a small parameter at the derivative and with a matrix pencil are obtained for various degeneracy orders of the structure matrix.