Block-diagonalisation of Matrices and Operators

In this short note we deal with a constructive scheme to decompose a continuous family of matrices A(ρ) asymptotically as ρ → 0 into blocks corresponding to groups of eigenvalues of the limit matrix A(0). We also discuss the extension of the scheme to matrix families depending upon additional parameters and operators on Hilbert spaces. 1. Matrix theory 1.1. Preliminaries. We first recall some well-known facts about matrix equations of Sylvester type and their solution. Let A, A ∈ C be two matrices with Re specA > 0, Re specA < 0. (1) Then a solution to the Sylvester equation AX −XA = B (2) for a given right hand side B ∈ C can be represented by the integral X = ∫ ∞ 0 e + Be − dt. (3) Indeed, by assumption (1) we know that there exists a constant c > 0 such that the matrix exponentials satisfy ‖e + ‖ ‖e − ‖ . e and the integral converges exponentially. Furthermore, plugging (3) into (2) immediately yields AX −XA = ∫ ∞ 0 ( Ae + Be − − e + Be − A )