369 HW7
1. Let v1 and v2 be eigenvectors of A corresponding to eigenvalues λ1 and λ2 .
(i) Show that if λ1 6= λ2 then v1 and v2 are linearly independent.
(ii) Show that if λ1 6= λ2 and A is symmetric, then v1 and v2 are orthogonal.1
2. Suppose that A is a matrix with characteristic polynomial pA (λ).
(i) Suppose that pA (λ) = (λ − 2)(λ + 3)(λ + 1). Can we say whether A is diagonalizable?
(ii) Suppose that pA (λ) = (λ − 2)(λ + 3)2 . Can we say whether A is diagonalizable?
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3. Compute by hand the minimal polynomial of A =
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4. Let A be a matrix with characteristic polynomial pA (λ) = (λ − 1)2 (λ2 + 1)(λ + 2)(λ + 5)4 .
(i) Is A diagonalizable?
(ii) What are the possible dimensions of the eigenspaces corresponding to λ = 1, −2, −5?
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Hint: Remember, for an n × n matrix A and any two vectors u, v ∈ Rn , we have Au · v = u · At v.
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