Linearizations of rational matrices from general representations

Javier Pérez*, María C. Quintana

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


We construct a new family of linearizations of rational matrices R(λ) written in the general form R(λ)=D(λ)+C(λ)A(λ)−1B(λ), where D(λ), C(λ), B(λ) and A(λ) are polynomial matrices. Such representation always exists and is not unique. The new linearizations are constructed from linearizations of the polynomial matrices D(λ) and A(λ), where each of them can be represented in terms of any polynomial basis. In addition, we show how to recover eigenvectors, when R(λ) is regular, and minimal bases and minimal indices, when R(λ) is singular, from those of their linearizations in this family.

Original languageEnglish
Pages (from-to)89-126
Number of pages38
JournalLinear Algebra and Its Applications
Publication statusPublished - 15 Aug 2022
MoE publication typeA1 Journal article-refereed


  • Block minimal bases pencil
  • Grade
  • Linearization at infinity
  • Linearization in a set
  • Rational eigenvalue problem
  • Rational matrix
  • Recovery of eigenvectors
  • Recovery of minimal bases
  • Recovery of minimal indices


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