### Abstract

We develop a noncommutative analogue of the spectral decomposition with the quasideterminant defined by I. Gelfand and V. Retakh. In this theory, by introducing a noncommutative Lagrange interpolating polynomial and combining a noncommutative Cayley-Hamilton's theorem and an identity given by a Vandermonde-like quasideterminant, we can systematically calculate a function of a matrix even if it has noncommutative entries. As examples, the noncommutative spectral decomposition and the exponential matrices of a quaternionic matrix and of a matrix with entries being harmonic oscillators are given.

Original language | English |
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Pages (from-to) | 2141-2158 |

Number of pages | 18 |

Journal | Advances in Mathematics |

Volume | 217 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2008 Mar 20 |

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### Keywords

- Noncommutative
- Quasideterminant
- Spectral decomposition

### ASJC Scopus subject areas

- Mathematics(all)