Suppose a $n\times n$ matrix $\mathbf{A}$ has $n$ independent eigenvectors $\mathbf{v}_1,...,\mathbf{v}_n$ with corresponding eigenvalues $\lambda_1,...,\lambda_n$. Then the general solution to $\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}$ is \begin{equation} \mathbf{x}(t)=\sum_{k=1}^n c_k\mathbf{v}_ke^{\lambda_k t} \end{equation} where $c_1,...,c_n$ are arbitrary constants.
In the case of an initial value problem with $\mathbf{x}(0)=\mathbf{x}^0$, is there anything that can be said about the constants $c_1,...,c_n$ apart from the fact that they need to satisfy $$ \sum_{k=1}^n c_k\mathbf{v}_k=\mathbf{x}^0 $$ My thinking is that, since the eigenvectors are not unique the multiples $c_k\mathbf{v}_k$ are also eigenvectors of $\mathbf{A}$ for the same eigenvalues, thus nothing else can be said about those. Is this correct?