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Problem 2. The eigenvalues of the following matrix are 5,0, -1. Find the associated eigenvectors manually, and then use those to find an orthogonal diagonalization. 1 2 A = 2 4 0 You may use Matlab or a calculator to do any difficult computations, but do not use it to find the eigenvectors or orthogonal diagonalization directly. (b) Use you answer from part (a) to find the spectral decomposition of A. You may use Matlab to do the spectral decomposition computations, but not to find the spectral decom- position directly. Write your answer in the form described in Problem (1b), except now it will be in the form AlM1 + A2M2 + AsMa for some M1, Me, and Ms.5. (a) State the Spectral Theorem for Symmetric matrices. (b) Write out the spectral factorization of 5 2 A = 2 2 (c) Using the change of variables x = Qy, where Q is an appropriate orthogonal matric, express the quadratic form 5r, + 4r142 + 21% in terms of y and y2. Describe and sketch the graph of the equation 5r, + 4712 + 2x2 = 1 in the T1, 12 plane. (d) Find the spectral decomposition of A.5. (a) State the Spectral Theorem for Symmetric matrices, (b) Write out the spectral factorization of A = 2 2 (c) Using the change of variables x = Qy, where Q is an appropriate orthogonal matric, express the quadratic form 5r; + 4riz, + 2x; in terms of y, and ya. Describe and sketch the graph of the equation 5r + 4riz, + 2r, = 1 in the $1, 12 plane. (d) Find the spectral decomposition of A.Problem 2. The eigenvalues of the following matrix are 5,0, -1. Find the associated eigenvectors manually, and then use those to find an orthogonal diagonalization. A = 0 0 You may use Matlab or a calculator to do any difficult computations, but do not use it to find the eigenvectors or orthogonal diagonalization directly. (b) Use you answer from part (a) to find the spectral decomposition of A. You may use Matlab to do the spectral decomposition computations, but not to find the spectral decom- position directly. Write your answer in the form described in Problem (1b), except now it will be in the form AM, + AM2 + AsMa for some M1, M2, and MS