Are all exponential functions eigensignals of LTI systems?
Complex exponentials are eigenfunctions of LTI systems because they are eigenfunctions of the convolution operator:
$$\begin{align}e^{j\omega_0t}\star h(t)&amp;=\int_{-\infty}^{\infty}h(\tau)e^{j\omega_0(t-\tau)}d\tau\\&amp;=e^{j\omega_0t}\int_{-\infty}^{\infty}h(\tau)e^{-j\omega_0\tau}d\tau\\&amp;=e^{j\omega_0t}H(j\omega_0)\end{align}\tag{1}$$
where $h(t)$ is the system's impulse response and $H(j\omega_0)$ is its frequency response (evaluated at $\omega=\omega_0$).
Now try to prove in the same way that the convolution of an impulse response $h(t)$ with a real-valued exponential results in a scaled version of that exponential. Then draw your conclusions.