Final Answer:
An eigenfunction (eigenvector) of Pₑ is a nonzero function f for which Pₑ(f) = λf, where λ is a real number. The eigenvalues corresponding to nonzero eigenvectors are the solutions to the equation Pₑ(f) = λf.
Step-by-step explanation:
Eigenfunctions and eigenvectors play a crucial role in linear algebra, and their application extends to operators like Pₑ. For a function f to be an eigenfunction of Pₑ, it must satisfy the equation Pₑ(f) = λf, where Pₑ represents a linear operator and λ is a real number. In mathematical terms, this can be expressed as Pₑ(f) - λf = 0. Nonzero functions satisfying this equation are the eigenfunctions, and the corresponding real numbers λ are the eigenvalues.
Consider the operator Pₑ acting on a function f(x). The equation Pₑ(f) = λf implies that applying Pₑ to f results in a scalar multiple of f. The nonzero eigenvalues λ correspond to the scaling factor by which the function is stretched or compressed under the action of Pₑ. The significance of eigenvalues lies in understanding how the operator Pₑ transforms functions and identifying the special functions that remain essentially unchanged, up to scaling, when operated on by Pₑ.
In summary, an eigenfunction of Pₑ is a function that, when operated on by Pₑ, is merely scaled by a real number λ. The eigenvalues represent the scaling factors, and solving the equation Pₑ(f) = λf reveals the specific functions and corresponding eigenvalues that satisfy this eigenvalue problem.