The cosine of a matrix is defined like eA , by copying the series for cos t: 1 2 1 4 cos

Chapter 6, Problem 31

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The cosine of a matrix is defined like eA , by copying the series for cos t: 1 2 1 4 cos t = 1 - - t + - t 2! 4! 1 2 1 4 cos A = I - - A + - A - ... 2! 4! (a) If Ax = AX, multiply each term times x to find the eigenvalue of cos A. (b) Find the eigenvalues of A = [= =] with eigenvectors (1, 1) and (1, -1). From the eigenvalues and eigenvectors of cos A, find that matrix C = cos A. (c) The second derivative of cos(At) is _A2 cos(At). d 2u u(t) = cos(At) u(O) solves dt2 = -A2 u starting from u' (0) = O. Construct u(t) = cos(At) u(O) by the usual three steps for that specific A: 1. Expand u(O) = (4,2) = CIX I + C2X2 in the eigenvectors. 2. Multiply those eigenvectors by and (instead of eAt). 3. Add up the solution u(t) = CI Xl + C2 X2.