Indeed lew us dene new coordinates y1 yn by setting x1

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Unformatted text preview: the eigenvalues of the symmetric matrix A. c. Find an orthonormal basis for R 3 consisting of eigenvectors of A. d. Find an orthogonal matrix B such that B −1 AB is diagonal. e. Find the general solution to the matrix differential equation d2 x = Ax. dt2 f. Find the solution to the initial value problem 1 2 dx = Ax, x(0) = 2 , dt2 0 dx (0) = 0. dt 2.4.3.a. Find the eigenvalues of the symmetric matrix −2 1 0 0 1 −2 1 0 . A= 0 1 −2 1 0 0 1 −2 b. What are the frequencies of oscillation of a mechanical system which is governed by the matrix differential equation d2 x = Ax? dt2 2.5 Mechanical systems with many degrees of freedom* Using an approach similar to the one used in the preceding section, we can consider more complicated systems consisting of many masses and springs. For 53 Figure 2.7: A circular array of carts and springs. example, we could consider the box spring underlying the mattress in a bed. Although such a box spring contains hundreds of individual springs, and hence the matrix A in the corresponding dyn...
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