The [[discrete-time complex exponential signal]] is an [[eigenfunction of a system|eigenfunction]] of a [[discrete-time LTI system]]:
$
x(n)=z^{n} \rightarrow y(n) = H(z)z^{n}
$
where $z\in \mathbb{C}$. If we know the [[impulse response]] of the LTI system, $h(n)$, we can use the [[convolution sum]] to compute the output signal:
$
y(n) =
\sum_{k=-\infty}^{+\infty} h(k) z^{(n-k)}
$
that is:
$
y(n)=
z^{n} \underbrace{\sum_{k=-\infty}^{+\infty} h(k) z^{-k}}_{H(z)}$
that is:
$y(n) = H(z) z^{n}$
where $H(z)$ is the eigenvalue associated with the eigenfunction $z^{n}$.