Title: 9. Least Common Multiples and Greatest Common Divisors of Matrix Polynomials
Abstract: Previous chapter Next chapter Classics in Applied Mathematics Matrix Polynomials9. Least Common Multiples and Greatest Common Divisors of Matrix Polynomialspp.231 - 251Chapter DOI:https://doi.org/10.1137/1.9780898719024.ch9PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt Let L1 (λ) ,… , Ls (λ) be a finite set of matrix polynomials. A matrix polynomial N(λ) is called a (left) common multiple of L1 (λ) ,… , Ls (λ) if Li (λ) is a right divisor of N (λ) ,i=1,… ,s . A common multiple N(λ) is called a least common multiple (1.c.m.) of L1 (λ) ,… , Ls (λ) 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 L1 (λ) ,… , Ls (λ) if D(λ) is a right divisor of every Li (λ) ; D(λ) is a greatest common divisor (g.c.d.) if D(λ) is a common divisor of L1 (λ) ,… , Ls (λ) 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 L1 (λ) ,… , Ls (λ) . 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. Previous chapter Next chapter RelatedDetails Published:2009ISBN:978-0-89871-681-8eISBN:978-0-89871-902-4 https://doi.org/10.1137/1.9780898719024Book Series Name:Classics in Applied MathematicsBook Code:CL58Book Pages:xxvii + 396Key words:spectral theory of matrix polynomials and applications
Publication Year: 2009
Publication Date: 2009-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
Access and Citation
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot