Suppose that A is a square matrix. Prove that a single vector may not be an eigenvector of A for two different eigenvalues

Suppose that the vector x≠ 0 is an eigenvector of A for the two eigenvalues λ and ρ, where λ ≠ρ. Then λ − ρ ≠0, and we also have

0 = Ax − Ax                                                     Property AIC

= λx − ρx              Definition EEM

= (λ − ρ)x     Property DSAC

By Theorem SMEZV,either λ − ρ = 0 or x = 0, which are both contradictions.