Geodesic
On similarity of an arbitrary matrix to a block diagonal matrix
Filomat, Tome 35 (2021) no. 4, p. 1205

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DOI

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 Dˆ = diag (A1,A2, ...,Am). I.e. there is an invertible matrix T, such that T−1AT = Dˆ. 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
DOI : 10.2298/FIL2104205G
Classification : 15A04, 15A42, 15A18
Keywords: matrices, similarity, condition number, operator functions, matrix function, resolvent: spectrum perturbation 
Michael Gil. On similarity of an arbitrary matrix to a block diagonal matrix. Filomat, Tome 35 (2021) no. 4, p. 1205 . doi: 10.2298/FIL2104205G
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     title = {On similarity of an arbitrary matrix to a block diagonal matrix},
     journal = {Filomat},
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     year = {2021},
     volume = {35},
     number = {4},
     doi = {10.2298/FIL2104205G},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2104205G/}
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