Remarks

• The proof shows that the period of the shift register is in fact the smallest positive number m such that A^m = I. This number is called the order of the matrix A. This number can be infinite. But if R is a finite field and A is an invertible matrix, then it is finite.
• The period of the shift register is not necessarily the period of every sequence that can generate. In fact, it depends on the initial state the shift register is in, as can be seen in one of the examples.