What conditions on $X$ and $W$ ensures the vectors $v_i = [\sum^{\infty}_{k=0} ( X \lambda_i )^k ]W$ are linearly independent? Let $W$ be a $n$ dimensional real vector. Let $\lambda_i$ be a list of $m$ distinct complex numbers. Let $X$ be a real $n \times n$ matrix whose spectral radius $\rho(X)$ satisfies $\rho(X) \lambda_i < 1$ for all $i$.
What conditions can I impose on $X$ and $W$ to ensure that the vectors
\begin{align*}
v_i = \bigg[\sum^{\infty}_{k=0} ( X \lambda_i )^k \bigg]W
\end{align*}
are linearly independent?
 A: As I state in my comment on the question, we require minimally that $X$ is non-derogatory and that $W$ is an $X$-cyclic vector.
With that all stated, we note that if the sum converges, it must satisfy
$$
\sum^{\infty}_{k=0} ( X \lambda_i )^k = (I  - \lambda_i X)^{-1}.
$$
Now, there exist coefficients $a^i_k$ such that
$$
(I  - \lambda_i X)^{-1} = a^i_0 I + a^i_1 X + \cdots + a^i_{n-1}X^{n-1}.
$$
Let $v_i = (a_0^i,a_1^i,\dots,a_{n-1}^i) \in \Bbb C^n$. Because the powers $I,X,\dots,X^{n-1}$ are linearly independent, we find that the matrix summation $\sum^{\infty}_{k=0} ( X \lambda_i )^k$ are linearly independent if and only if the vectors $v_i$ are linearly independent.
With that said: given that the matrices $\sum^{\infty}_{k=0} ( X \lambda_i )^k$ are linearly independent, it is not clear how one should choose $W$ to ensure that the vectors $\left[\sum^{\infty}_{k=0} ( X \lambda_i )^k\right]W$ are linearly independent.

Incorrect work:
Now, suppose that $X$ has characteristic polynomial $p(x) = \sum_{k=0}^n a_k x^k$, with $a_n = 1$. We note that
$$
a_0 I + a_1 \left[\frac{I - (I - \lambda_i X)}{\lambda_i}\right] + \cdots + a_{n-1}\left[\frac{I - (I - \lambda_i X)}{\lambda_i}\right]^{n-1} + \left[\frac{I - (I - \lambda_i X)}{\lambda_i}\right]^n = 0.
$$
Now, let $q_i(x) = \lambda_i^n p(\frac{1 - x}{\lambda_i}) = \sum_{k=0}^n a^i_k x^k$, with $a^i_n = 1$. We have
$$
a_0^i I + a_1^i (I - \lambda_i X) + \cdots + a_{n-1}^i(I - \lambda_i X)^{n-1} + (I - \lambda_i X)^n = 0 \implies\\
(I - \lambda_i X) \cdot \frac{a_1^i I + \cdots + a_{n-1}^i(I - \lambda_i X)^{n-2} + (I - \lambda_i X)^{n-1}}{-a_0^i}  = I \implies\\
(I - \lambda_i X)^{-1} = \frac{a_1^i I + \cdots + a_{n-1}^i(I - \lambda_i X)^{n-2} + (I - \lambda_i X)^{n-1}}{-a_0^i}.
$$
Now, let $v_i = (a_1^i,\dots,a_{n-1}^i,1) \in \Bbb C^n$. Because the powers $I,X,\dots,X^{n-1}$ are linearly independent, we find that the matrix summation $\sum^{\infty}_{k=0} ( X \lambda_i )^k$ are linearly indepednent if and only if the vectors $v_i$ are linearly independent.
