9. Least Common Multiples and Greatest Common Divisors of Matrix Polynomials

Let be a finite set of matrix polynomials. A matrix polynomial N(λ) is called a (left) common multiple of if is a right divisor of . A common multiple N(λ) is called a least common multiple (1.c.m.) of if N(λ) is a right divisor of every other common multiple. The notions of a common divisor and a greatest common divisor are given in an analogous way : namely, a matrix polynomial D(λ) is a (right) common divisor of if D(λ) is a right divisor of every ; D(λ) is a greatest common divisor (g.c.d.) if D(λ) is a common divisor of and every other common divisor is in turn a right divisor of D(λ). It will transpire that the spectral theory of matrix polynomials, as developed in the earlier chapters, provides the appropriate machinery for solving problems concerning l.c.m. and g.c.d. In particular, we give in this chapter an explicit construction of the l.c.m. and g.c.d. of a finite family of matrix polynomials . The construction will be given in terms of both the spectral data of the family and their coefficients. The discussion in terms of coefficient matrices is based on the use of Vandermonde and resultant matrices, which will be introduced later in the chapter.