Let $A_n$ be a sequence of $d \times d$ matrices converging to a matrix $A$, all invertible and diagonalizable. We can define the Lyapunov spectrum of the corresponding dynamical system: $$ \chi = \{ \lim_{n \to \infty} \frac1n \log\||A^{(n)}v\|| \mid\ v \in \mathbb{R^d} \}$$

where $A^{(n)} = A_n A_{n-1} A_{n-2} \cdots A_1$. Is it the case, in general, that $\chi$ equals the Lyapunov spectrum of the matrix $A$? If not, can we make this true by assuming that the sequence converges fast enough (for some notion of "fast enough")?