Since the state transformation of the shift register contents is a linear operation, a matrix equation describing it can be given. Continuing with the example of the 10-long generator, if we define and by:
then the action of the shift register can be readily described by the equation
where is the entire vector after n time steps. If the vector has been given some initial set of values, then we have,
etc., and in general,
As an aside, note that , or , the identity matrix.