# qualitative results using discrete Putzer algorithm

I am studing for an exam in difference equations...the following is an exarcise from S.N.Elaydi's book.

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Define $\displaystyle{ u_1 (n) = \lambda _1 ^n , \quad u_j (n) = \sum_{i=0}^{n-1} \lambda_j ^{ n-1-i} u_{j-1} (i) \quad j=2,3, \cdots ,k }$ ,where $\lambda_j \quad j=1,2,\cdots, k$ are the eigenvalues of a matrix $A$.

Let $\displaystyle{ \rho(A)= \max \{ | \lambda | : \lambda \text{ is an eigenvalue of A } \} }$. Suppose that $\rho (A) =\rho_0 < \beta$.

(a) Show that $\displaystyle{ | u_j (n) | \leq \frac{ \beta ^n}{ \beta - \rho_0} \quad j=1,2,\cdots ,k}$

(b) Show that if $\rho_0 <1$ then $u_j (n) \to 0$ as $n \to \infty$. Concuding with $A^n \to 0$ as $n \to \infty$.

(c)If $\displaystyle{ \alpha < \min \{ | \lambda | : \lambda \text{ is an eigenvalue of A } \} }$ , establish a lower bound for $| u_j (n) |$.

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I think I have a solution for (b) if I assume that $\beta <1$. Can I assume this?

Can someone help with (a) and (c)?