Suppose that λ and ρ are two different eigenvalues of the square matrix A. Prove that the intersection of the eigenspaces for these two eigenvalues is trivial

This problem asks you to prove that two sets are equal, so use Definition SE.

First show that {0} ⊆ E_A (λ) ∩ E_A (ρ). Choose x ∈ {0}. Then x = 0. Eigenspaces are subspaces (Theo- rem EMS), so both E_A (λ) and E_A (ρ) contain the zero vector, and therefore x ∈ E_A (λ)∩E_A (ρ) (Definition SI).

To show that  E_A (λ)   ∩ E_A (ρ)     ⊆0 , suppose that x  ∈ E_A (λ)∩E_A (ρ). Then x is an eigenvector of A for both λ and ρ (Definition SI) and so

So x = 0, and trivially, x ∈ {0}.