I noticed a strong similarity in the proofs of the linear independence of characters (homomorphisms from a given group $G$ to units of $\mathbb C$) and of eigenvectors with different eigenvalues).
However, I can't seem to find an obvious relationship between them (either a character as a kind of eigenvector or visa versa). I have found one relationship, but it feels unnatural. I understand that a character can be identified with a vector in $\mathbb C^{G}$, but I don't see an obvious way of defining an eigenvalue which is distinct for distinct characters. I do understand that if you define a matrix such that the entry corresponding to the elements of the group a and b is $f(ab^{-1})$ where $f$ is some character, then every character will be an eigenvector and if we consider linear combinations of such matrices, we should be able to find such a matrix, but this seems to rely on more complicated facts and may even be self referential.
Also, similar to the above is making a Cayley graph out of the group an using the adjacency matrix, but again, this feels unnatural unless I'm misunderstanding it.
Thanks for any advice you can give. I feel like there's something obvious that I'm missing.