A substitute of l’Hospital’s rule for matrices

Abstract:

In this paper the following limit theorem is obtained: If $A$ and $B$ are $(n,n)$-matrices with ${\text {rank}}({A^T},{B^T}) = n,\;{A^T}B = {B^T}A$, then $A{(A + SB)^{ - 1}}S \to 0$ as $S \to 0 + ,\;{\text {i}}{\text {.e}}{\text {.}}\;{\text {S}} \to {\text {0}}$ where $S$ is symmetric and positive definite. Some applications of this result are given to linear algebra (the behavior of ${(A + \lambda B)^{ - 1}}$ as $\lambda \to 0)$ and to differential equations (the asymptotic behavior of Hamiltonian systems and of selfadjoint differential equations of even order).