Eigen Function of Linear Time Invariant (LTI) System Whenever I try to find the derivation of eigen signal of a LTI system, they always start with assumption that the input signal is complex exponential. Is there any other way to prove that this is the case starting from the basic equation of LTI system
$$\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau = \lambda x(t)$$
where $\lambda$ is the eigenvalue of the eigen signal
You can use the Convolution Theorem to show that $ \int_{-\infty}^{\infty} h \left( \tau \right) x \left( t - \tau \right) d \tau $ is equivalent to Element Wise multiplication in the Frequency Domain.  

Hence for the case $ x \left( t \right) = A {e}^{s t} $ where $ s \in \mathbb{C} $ the output is just a multiplication by scalar of the input (The value of the System Transfer Function at the Delta point of the input).

See Eigen Functions of LTI Systems.