Solutions of a class of singular linear systems of difference equations. Part 1

ABSTRACT

We extend results of Campbell, Meyer, Jr. and Rose of applications of the Drazine inverse to linear systems of differential equations with singular constant coefficients to solutions of linear systems of difference equations 𝐴 𝑥𝑛+1+𝐵 𝑥𝑛= 𝑓𝑛 ,𝑛≥0 when 𝐴 and 𝐵 are 𝑚×𝑚 complex matrices and may both singular, under conditions that 𝑟𝑎𝑛𝑘(𝐴)=1 and trace of 𝐴 is not equal zero. 𝑓𝑛 is an arbitrary function in ℂ𝑚, and 𝑥𝑛∈ℂ𝑚. We give a new closed form for all solutions of those systems when they are tractable, using the theory of the Drazin inverse, and a matrix 𝐾∈ℂ𝑚×𝑚.