• a square polynomial matrix Q(s) is unimodular iff det Q(s) is constant.
• Let Q(s) be a (p×m) polynomial matrix. Then the normal rank of Q(s), denoted normalrank (Q(s)), is the maximally
possible rank of Q(s) for at least one sÎ C.
Example:
Q(s) has normal rank 2 since
rank Q(2) = 2. However, Q(0) has rank 1.
• Smith
form: Let
P(s) be any polynomial matrix,
then there exist unimodular matrices U(s), V(s) Î R[s] such that
•
where gi(s) divides gi+1(s) and r is the normal rank of P(s).
S(s) is called the Smith form of P(s).