On similarity of an arbitrary matrix to a block diagonal matrix

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Let an n × n-matrix A have m < n (m ≥ 2) different eigenvalues λj of the algebraic multiplicity µj (j = 1, …, m). It is proved that there are µj × µj-matrices Aj, each of which has a unique eigenvalue λj, such that A is similar to the block-diagonal matrix ˆD = diag (A1, A2, …, Am). I.e. there is an invertible matrix T, such that T−1AT = ˆD. Besides, a sharp bound for the number κT:= ‖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 languageEnglish
Pages (from-to)1205-1214
Number of pages10
Issue number4
StatePublished - 1 Jan 2021


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

ASJC Scopus subject areas

  • General Mathematics


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