a6 - 1 1 1 1 1 1-1-1 1-1 1-1 1-1-1 1 . 4. (Bonus Question)...

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McGill University Math 236: Algebra 2 Assignment 6: due Monday, April 10, 2006 1. (a) Find the spectral decomposition of the matrix A = 3 1 1 0 1 3 0 1 1 0 3 1 0 1 1 3 Hint: By inspection 3 and 5 are eigenvalues of A . Explain why this is the case. (b) Using (a), solve the initial value problem i dX dt = Ax, X (0) = 1 1 1 1 . (c) Using (a), find the maximum and minimum values of the quadratic form q ( X ) = 3 x 2 1 + 2 x 1 x 2 + 2 x 1 x 3 + 3 x 2 2 + 2 x 2 x 4 + 3 x 2 3 + 2 x 3 x 4 + 3 x 2 4 subject to the constraint x 2 1 + x 2 2 + x 2 3 + x 2 4 = 1 and find where these maxima and minima are attained. 2. Find the Jordan canonical form of the matrix A = 1 - 1 0 0 1 0 1 1 - 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 and find an invertible matrix such that P - 1 AP is in Jordan canonical form. 3. (Bonus Question) Find the spectral decomposition of the matrix A =
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Unformatted text preview: 1 1 1 1 1 1-1-1 1-1 1-1 1-1-1 1 . 4. (Bonus Question) (a) Prove or give a counterexample: the product of any two self-adjoint operators on an inner-product space is self-adjoint. (b) If T is a normal operator on a nite-dimensional complex inner product space such that T 9 = T 8 prove that T is self-adjoint and T 2 = T . 5. (Bonus Question) (a) Let T be a linear operator on a vector space V such that dim(Im( T )) = n . Prove that T has at most n + 1 distinct eigenvalues. (b) If T is a linear operator on a vector space V such that any vector of V is an eigenvector of T prove that T is a scalar multiple of the identity operator....
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