Open AccessOn similarity of an arbitrary matrix to a block diagonal matrix
Author(s) -
Michael Gil
Publication year - 2021
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil2104205g
Subject(s) - mathematics , combinatorics , block matrix , invertible matrix , eigenvalues and eigenvectors , matrix (chemical analysis) , algebraic number , diagonal matrix , matrix norm , block (permutation group theory) , diagonal , pure mathematics , mathematical analysis , geometry , physics , materials science , quantum mechanics , composite material
Let an n x n -matrix A have m < n (m ? 2) different eigenvalues ?j of the algebraic multiplicity ?j (j = 1,..., m). It is proved that there are ?j x ?j-matrices Aj, each of which has a unique eigenvalue ?j, such that A is similar to the block-diagonal matrix ?D = diag (A1,A2,..., Am). I.e. there is an invertible matrix T, such that T-1AT = ?D. Besides, a sharp bound for the number kT := ||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.