On Similarity of an Arbitrary Matrix to a Block Diagonal Matrix

Abstract

Let an $n\by n$ -matrix $A$ have 
$m< n$ $(m\ge 2)$ different eigenvalues 
$\la_j$ of the algebraic multiplicity $\mu_j$  $(j=1, ..., m)$.
It is proved that there are $\mu_j\by \mu_j$-matrices $A_j,$ each of which has a unique eigenvalue $\la_j$,  such
that $A$ is similar to the block-diagonal matrix $\hat D=\diag\;(A_1, A_2, ..., A_m)$. I.e.    there
 is  an  invertible  matrix $T$, such that
$T\mi A T=\hat D$. Besides, a sharp bound for the
 number $\ka_T:=\|T\|\|T\mi\|$ 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.