Let \(A : {\mathbb R} \to {\mathbb C}^{m\times m}\) be a continuous matrix-valued function depending upon a real/complex parameter \(\rho\) such that the family of matrices has the asymptotic expansion \(A(\rho) = A_0 + \rho A_1 + \rho^2 A_2 +\cdots+\rho^N A_N + O(\rho^{N+1})\), \(\rho\to 0\), \(N\in\mathbb N\), and \(A_0\) is non-degenerate (i.e., \(A_0\) has \(m\) distinct eigenvalues). The authors start from the existence of uniformly bounded families of invertible matrices \(M(\rho)\) with uniformly bounded inverse having asymptotic expansions as \(\rho\to 0\) and satisfying \(A(\rho)M(\rho) -M(\rho)\Lambda(\rho) = O(\rho^N)\), \(N\in \mathbb N\), for a diagonal matrix \(\Lambda(\rho)\). They show how to construct the diagonaliser \(M(\rho)\) using a recursion scheme and how the aforementioned uniform bounds and the asymptotic expansion of \(\Lambda(\rho)\) arise naturally within the construction. The main objective of this note is to discuss how to generalise this scheme to degenerate matrix functions and to replace the assumption of non-degeneracy by weaker assumptions.
Applications in different frameworks are also discussed and references to further applications given.