Let \(\{M_ n\}_{n \geq 1}\) be a sequence of non-singular \(k \times k\) matrices which converges to a block-diagonal matrix \(M = \text{diag }(R,S)\), where \(R\) and \(S\) are square matrices. The author finds sufficient conditions for the existence of a sequence \(\{F_ n\}\) of non-singular \(k \times k\) matrices such that \[ F_{n + 1}^{-1} M_ n F_ n = \text{diag }(\widetilde{R}_ n,\widetilde {S}_ n) \] where \(\widetilde{R}_ n\) and \(\widetilde{S}_ n\) have the same dimensions as \(R\) and \(S\) respectively, and \(\widetilde{R}_ n \to R\), \(\widetilde{S}_ n \to S\) and \(F_ n \to I\), the \(k \times k\) identity matrix. The speed of convergence is quantified. A similar result is also derived for the case \(M = \text{diag}(R_ 1,R_ 2, \dots,R_ L)\) where all \(R_ m\) are square blocks. The author applies these results to find the asymptotic behavior of solutions \(\{x_ n\}\) of the matrix recurrence relation \[ x_{n + 1} = M_ n x_ n \quad \text{for} \quad n = 1,2,3,\dots \] where \(\{x_ n\}\) are either vectors or \(k \times k\) matrices. This leads among other things to a matrix version of the Poincaré-Perron theorem.