Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems Articles uri icon

publication date

  • November 2020

start page

  • 441

end page

  • 475


  • 604

International Standard Serial Number (ISSN)

  • 0024-3795

Electronic International Standard Serial Number (EISSN)

  • 1873-1856


  • This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. This new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2020 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.


  • Mathematics


  • rational matrix; rational eigenvalue problem; nonlinear eigenvalue problem; linearization; polynomial system matrix; rational approximation; block full rank pencils