Construction of a fundamental matrix solution at a singular point of the first kind by means of the SN decomposition of matrices Original Research Article

It is known that any matrix can be decomposed into a diagonalizable part and a nilpotent part. We call this the SN decomposition. We can derive the SN decomposition quite easily with a computer. Generalizing the SN decomposition to particular matrices of infinite order, we explain basic steps of construction of a linear transformation which reduces a given system of linear meromorphic ordinary differential equations to a normal form at a singular point of the first kind. Some examples are given utilizing Mathematica. We also show that the same idea produces a block-diagonalization of a given system at a singular point of the second kind.