## Abstract

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 language | English |
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Pages (from-to) | 37-62 |

Number of pages | 26 |

Journal | Linear Algebra and Its Applications |

Volume | 684 |

DOIs | |

State | Published - 1 Mar 2024 |

### Bibliographical note

Publisher Copyright:© 2023 Elsevier Inc.

### Funding

The research of the first author was supported by ISF Grant No. 1994/20 . The research of the third and the forth authors was supported by ISF Grant No. 1092/22 .

Funders | Funder number |
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Israel Science Foundation | 1092/22, 1994/20 |

## Keywords

- Differentiators
- Integrators
- Matrices
- Polynomials