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Unformatted text preview: MATH 108B HW 3 SOLUTIONS RAHUL SHAH Problem 1. [ Â§ 7.11] Solution. Assume T is a normal operator on a complex vector space. Then, by the complex spectral theorem, we find that T has an orthonormal basis consisting of eigenvectors. Thus M ( T ) is a diagonal matrix (with respect to the given orthonormal basis of eigenvectors). Let exp Ä±Î¸ j be the jth diagonal entry of M ( T ). Let Î» j = exp Ä± Î¸ j 2 (choose one of the square roots  exists since z 2 c always has a root over the complex numbers for any c , a complex number). Let M ( T ) be a diagonal matrix with the jth diagonal entry given by Î» j and let S be the operator that has matrix M ( S ) with respect to the given orthonormal basis of eigenvectors of T . Notice that S 2 = T . Problem 2. [ Â§ 7.12] Solution. Let V = R 2 , and let T be the operator given by a 90 â—¦ rotation. Let Î± = 0 and let Î² = 1. Then Î± 2 < 4 Î² . Notice that T 2 = I and thus T 2 + Î±T + Î²I = 0, which is clearly not invertible. Problem 3. [ Â§ 7.13] Solution. Notice that for V , either a real or a complex vector space, and T a selfadjoint operator on T , T has an orthonormal basis consisting of eigenvectors (since every selfadjoint operator is normal, this also works for complex...
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 Winter '09
 Linear Algebra, Eigenvectors, Vectors, Vector Space, Hilbert space, Orthonormal basis, RAHUL SHAH

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