The eigenvalues of an operator are the only possible values because they represent the values for which the operator acts as a scalar multiplier on its corresponding eigenvector. In other words, these are the only values that the operator can "eigenvectorize" or "eigenfunctionize."
An operator's eigenvalues provide important information about its behavior and properties. They can indicate symmetry, stability, and other fundamental characteristics of the operator.
An operator is diagonalizable if and only if it has a full set of linearly independent eigenvectors. The eigenvalues are the entries on the diagonal of the corresponding diagonal matrix, hence the term "diagonalizability."
Yes, an operator can have complex eigenvalues. In fact, in many cases, the eigenvalues of an operator are complex numbers, especially in quantum mechanics and other areas of physics.
Eigenvalues and eigenvectors are used to solve systems of linear differential equations by transforming them into a diagonal form, where each equation can be solved independently. This approach is known as the method of eigenfunction expansion or the method of separation of variables.