On similarity of an arbitrary matrix to a block diagonal matrix

Let an n x -matrix A have m < (m ? 2) different eigenvalues ?j of the algebraic multiplicity (j = 1,..., m). It is proved that there are ?j-matrices Aj, each which has a unique eigenvalue ?j, such similar to block-diagonal matrix ?D diag (A1,A2,..., Am). I.e. invertible T, T-1AT ?D. Besides, sharp bound for number kT := ||T||||T-1|| derived. As applications these results we obtain norm estimates functions non-regular on convex hull spectra. These generalize and refine previously published results. In addition, new spectral variation matrices appropriate situations it refines well known bounds.