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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 Q1 AQ = D for each of the following matrices A ∈ M n × n ( F ), (a) A = ±2 1 3 ² for F = R . (Solution) Q = ±21 1 1 ² , D = ± 1 2 ² (b) A = ± i 1 2i ² for F = C . (Solution) Q = ± 1 1 1i1i ² , D = ± 11 ² (c) A = 3 1 1 2 4 211 1 for F = R . 1 (Solution) Q = 1 1 1 2111 , D = 2 2 4 [Hint] i Determine all the eigenvalues of A . ii For each eigenvalue λ of A , ﬁnd the set of eigenvectors corresponding 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 Q1 AQ = D . 2...
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 Spring '06
 Holtz
 Linear Algebra, Algebra, qV, similar matrices, Eλ

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