# When n is large we can consider this array of springs

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ωj = 2π k sin(πj/n) . m π Note that the component 1 1 e0 (a0 + b0 t) = 1 (a0 + b0 t) · 1...
View Full Document

## This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online