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Unformatted text preview: genspaces for A are two-dimensional,
except for the eigenspace corresponding to j = 0, and if n is even, to the
eigenspace corresponding to j = n/2. Thus in the special case where n is odd,
all the eigenspaces are two-dimensional except for the eigenspace with eigenvalue
λ0 = 0, which is one-dimensional.
If j = 0 and j = n/2, ej and en−j form a basis for the eigenspace corresponding to eigenvalue λj . It is not diﬃcult to verify that 1 cos(πj/n) 1 cos(2πj/n) (ej + en−j ) = 2 ·
cos((n − 1)πj/n)
and 0 sin(πj/n) i sin(2πj/n) (ej − en−j ) = 2 ·
sin((n − 1)πj/n)
57 form a real basis for this eigenspace. Let
−λj = 2 sin(πj/n). ωj = In the case where n is odd, we can write the general solution to our dynamical
system (2.17) as
(n−1)/2 x = e0 (a0 + b0 t) +
j =1 1
(ej + en−j )(aj cos ωj t + bj sin ωj t)
2 (n−1)/2 +
j =1 i
(ej − en−j )(cj cos ωj t + dj sin ωj t).
2 The motion of the system can be described as a superposition of several modes
of oscillation, the frequencies of oscillation being
2π k sin(πj/n)
π Note that the component 1
1 e0 (a0 + b0 t) = 1 (a0 + b0 t) ·
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This document was uploaded on 01/12/2014.
- Winter '14