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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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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