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Unformatted text preview: MATH 108B HW 3 SOLUTIONS RAHUL SHAH Problem 1. [ § 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 ıθ j be the j-th diagonal entry of M ( T ). Let λ j = exp ı θ 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 j-th diagonal entry given by λ 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. [ § 7.12] Solution. Let V = R 2 , and let T be the operator given by a 90 ◦ rotation. Let α = 0 and let β = 1. Then α 2 < 4 β . Notice that T 2 =- I and thus T 2 + αT + βI = 0, which is clearly not invertible. Problem 3. [ § 7.13] Solution. Notice that for V , either a real or a complex vector space, and T a self-adjoint operator on T , T has an orthonormal basis consisting of eigenvectors (since every self-adjoint operator is normal, this also works for complex...
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