On similarity of an arbitrary matrix to a block diagonal matrix

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 ˆD = diag (A₁, A₂, …, A_m). I.e. there is an invertible matrix T, such that T⁻¹AT = ˆD. Besides, a sharp bound for the number κ_T:= ‖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.