### 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 |
---|---|

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 |

Externally published | Yes |

### Keywords

- Noncommutative
- Quasideterminant
- Spectral decomposition

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Noncommutative spectral decomposition with quasideterminant.** / Suzuki, Tatsuo.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 217, no. 5, pp. 2141-2158. https://doi.org/10.1016/j.aim.2007.09.011

}

TY - JOUR

T1 - Noncommutative spectral decomposition with quasideterminant

AU - Suzuki, Tatsuo

PY - 2008/3/20

Y1 - 2008/3/20

N2 - 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.

AB - 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.

KW - Noncommutative

KW - Quasideterminant

KW - Spectral decomposition

UR - http://www.scopus.com/inward/record.url?scp=38849090531&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38849090531&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2007.09.011

DO - 10.1016/j.aim.2007.09.011

M3 - Article

AN - SCOPUS:38849090531

VL - 217

SP - 2141

EP - 2158

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 5

ER -