Decoupling of second-order linear systems by isospectral transformation

  • Daniel T. KawanoEmail author
  • Rubens G. SalsaJr.
  • Fai Ma


We consider the class of real second-order linear dynamical systems that admit real diagonal forms with the same eigenvalues and partial multiplicities. The nonzero leading coefficient is allowed to be singular, and the associated quadratic matrix polynomial is assumed to be regular. We present a method and algorithm for converting any such n-dimensional system into a set of n mutually independent second-, first-, and zeroth-order equations. The solutions of these two systems are related by a real, time-dependent, and nonlinear n-dimensional transformation. Explicit formulas for computing the \(2n \times 2n\) real and time-invariant equivalence transformation that enables this conversion are provided. This paper constitutes a complete solution to the problem of diagonalizing a second-order linear system while preserving its associated Jordan canonical form.


Second-order linear differential equations Quadratic matrix polynomials Diagonalization Isospectral systems 

Mathematics Subject Classification

15A22 34A30 70J10 



RGS was supported by a Science Without Borders fellowship from the CAPES Foundation (grant no. 99999.011952/2013-00).


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Daniel T. Kawano
    • 1
    Email author
  • Rubens G. SalsaJr.
    • 2
  • Fai Ma
    • 2
  1. 1.Department of Mechanical EngineeringRose-Hulman Institute of TechnologyTerre HauteUSA
  2. 2.Department of Mechanical EngineeringUniversity of California, BerkeleyBerkeleyUSA

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