## Abstract

Let an n × n-matrix A have m < n (m ≥ 2) different eigenvalues λ_{j} of the algebraic multiplicity µ_{j} (j = 1, …, m). It is proved that there are µ_{j} × µ_{j}-matrices A_{j}, each of which has a unique eigenvalue λ_{j}, such that A is similar to the block-diagonal matrix ˆD = diag (A_{1}, A_{2}, …, A_{m}). I.e. there is an invertible matrix T, such that T^{−1}AT = ˆD. Besides, a sharp bound for the number κ_{T}:= ‖T‖‖T^{−1}‖ is derived. As applications of these results we obtain norm estimates for matrix functions non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition, a new bound for the spectral variation of matrices is derived. In the appropriate situations it refines the well known bounds.

Original language | English |
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Pages (from-to) | 1205-1214 |

Number of pages | 10 |

Journal | Filomat |

Volume | 35 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 2021 |

## Keywords

- Condition number
- Matrices
- Matrix function
- Operator functions
- Resolvent: spectrum perturbation
- Similarity

## ASJC Scopus subject areas

- General Mathematics