When addressing people who do not have the necessary mathematical prerequisites to understand the technical details, it is imperative that you locate the point of contact between their daily lives and the topic at hand. When you are successful, then it becomes impossible for them to question the relevance of your talk.
I would open with the famous movie of the Tacoma bridge suffering from wind induced oscillations which end with violent structural collapsed. I would mention that the problem could have been predicted and avoided through computation of the eigenvalues for the stiffness matrix.
I would mention signal analysis and digitalization of sound and images, a topic where Fourier analysis is central.
Time permitting I would mention research into the fundamental properties of matter, i.e., quantum mechanics, the study of which hinges on our understanding of unbounded linear operators, i.e., an extension of your topic.
Instead of technical details, explain that these topics can not be accessed, understood and advanced further, if they do not posses the mathematical skills that you can teach them.