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