(Diagonal matrices)
a) Let B = diag (λ, μ), Show that, if λ ≠ μ, then the eigenvectors associated with λ are all nonzero vectors c(1, 0) and those associated with μ are all nonzero vectors c(0, 1); show that if λ = μ then the eigenvectors associated with λ are all nonzero vectors (v1, v2).
b) Let B = diag(λ1, λ2, λ3) and let e1 =(1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) (column vectors). Show that if λ1, λ2, λ3 are distinct, then for each λk the associated eigenvectors are the vectors cek for c ≠ 0; show that if λ1 = λ2 ≠ λ3, then the eigenvectors associated with λ1 are all nonzero vectors c1e1 + c2e2 and those associated with λ3 are all nonzero vectors ce3; show that if λ1 = λ2 = λ3, then the eigenvectors associated with λ1 are all nonzero vectors v = (v1, v2, v3).
c) Let B = diag (λ1…, λn). Show that the eigenvectors associated with the eigenvalue λk are all nonzero vectors v = (v1, ..., vn) such that vi = 0 for all i such that λ1, ≠ λk.
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