In my classical field theory classes I have an easy time understanding most topics. I think this is because I have a good conceptual understanding of the tools used. Lagrange equations and such. However, I find I am having much more trouble in my quantum classes. I cant get the same conceptual understanding. Anyone have any ideas how to fix this? Here are some questions I have that I think might help

What exactly is an operator, "an observable sure" but why do we say its a matrix. What does it mean if we have some operator with entries in the whole matrix. If we have entries only along the diagonal, we say its Hermitian and these entries are eigenvalues. Eigenvalues of what? It was my understanding if you took some matrix you find eigenvalues for that matrix, not that they were present in the matrix. Is a Hermitian matrix with eigenvalues in the diagonal just a matrix of eigenvalues of some other matrix?


Why do we represent functions as bras and kets. If we had some function y=x^2, is a bra all x entries and the ket all y entries in this function?


Retarded I know but can I get a quick summary of eigenvalue/vector. Like is a eigenstate a eigenvalue? ect


Any ways you use to help get an understanding of qm would be appreciated, ill likely post more question if I get answers, thanks.