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# quiz4 - V has a basis consisting of eigenvectors of...

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Quiz #4 1. Label the following statements as being true or false. (a) Any two eigenvectors are linearly independent. (Solution) False. Consider an constant multiple of an eigenvec- tor. (b) Similar matrices always have the same eigenvalues. (Solution) True. Similar matrices have the same characteristic polynomial. (c) Similar matrices always have the same eigenvectors. (Solution) False. Let A = Q - 1 BQ for some Q . Then every eigenvector of A is of the form Q - 1 v for some eigenvector of B . Similarly, every eigenvector of B is of the form Qv for some eigenvector of A . (d) A linear operator T on a ﬁnite-dimensional vector space is di- agonalizable if and only if the multiplicity of each eigenvalue λ equals the dimension of E λ . (Solution) False. In addition, the sum of all dimension of E λ ’s must be equal to the dimension of the vector space. (e) A linear operator T : V V on a ﬁnite dimensional vector space V is diagonalizable if and only if

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Unformatted text preview: V has a basis consisting of eigenvectors of T . (Solution) True. 2. Determine an invertible matrix Q and a diagonal matrix D such that Q-1 AQ = D for each of the following matrices A ∈ M n × n ( F ), (a) A = ±-2 1 3 ² for F = R . (Solution) Q = ±-2-1 1 1 ² , D = ± 1 2 ² (b) A = ± i 1 2-i ² for F = C . (Solution) Q = ± 1 1 1-i-1-i ² , D = ± 1-1 ² (c) A = 3 1 1 2 4 2-1-1 1 for F = R . 1 (Solution) Q = 1 1 1 2-1-1-1 , D = 2 2 4 [Hint] i Determine all the eigenvalues of A . ii For each eigenvalue λ of A , ﬁnd the set of eigenvectors corre-sponding to λ . iii Find a basis for F n consisting of eigenvectors of A . iv Determine an invertible matrix Q and a diagonal matrix D such that Q-1 AQ = D . 2...
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quiz4 - V has a basis consisting of eigenvectors of...

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