Strong Linearizations of Rational Matrices Articles uri icon

publication date

  • November 2018

start page

  • 1670

end page

  • 1700


  • 4


  • 39

International Standard Serial Number (ISSN)

  • 0895-4798

Electronic International Standard Serial Number (EISSN)

  • 1095-7162


  • This paper defines for the first time strong linearizations of arbitrary rational matrices, studies in depth properties and characterizations of such linear matrix pencils, and develops infinitely many examples of strong linearizations that can be explicitly and easily constructed from a minimal state-space realization of the strictly proper part of the considered rational matrix and the coefficients of the polynomial part. As a consequence, the results in this paper establish a rigorous foundation for the numerical computation of the complete structure of zeros and poles, both finite and at infinity, of any rational matrix by applying any well-known backward stable algorithm for generalized eigenvalue problems to any of the strong linearizations constructed in this work.


  • Mathematics


  • linearization, minimal polynomial system matrix, nonlinear eigenvalue problem; rational matrix, strong block minimal bases linearization, strong linearization