Integrability of matrices

S. Danielyan, A. Guterman, E. Kreines, F. Pakovich

Research output: Contribution to journalArticlepeer-review


The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable matrices, however general problem remained open. In this paper, we present a full solution of the integrability problem. Namely, we provide necessary and sufficient conditions for a given matrix to be integrable in terms of its characteristic polynomial. Furthermore, we find necessary and sufficient conditions for the existence of integrable and non-integrable matrices with given geometric multiplicities of eigenvalues. Our approach relies on properties of some special classes of polynomials, namely, Shabat polynomials and conservative polynomials, arising in number theory and dynamics.

Original languageEnglish
Pages (from-to)37-62
Number of pages26
JournalLinear Algebra and Its Applications
StatePublished - 1 Mar 2024


  • Differentiators
  • Integrators
  • Matrices
  • Polynomials

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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