8 a linear array of carts and springs by expanding

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Unformatted text preview: r for T corresponding to the eigenvalue λ, the components of x must satisfy the vector equation T x = λx. In terms of components, the vector equation becomes x2 = λx1 , x3 = λx2 , . . . , xn = λxn−1 , x1 = λxn . (2.19) Thus x3 = λ2 x1 , x4 = λ3 x1 , and so forth, yielding finally the equation x1 = λn x1 . Similarly, x2 = λn x2 , ... , xn = λn xn . Since at least one of the xi ’s is nonzero, we must have λn = 1. (2.20) This equation is easily solved via Euler’s formula: 1 = e2πi ⇒ (e2πi/n )n = 1, and similarly [(e2πi/n )j ]n = 1, for 0 ≤ j ≤ n − 1. Thus the solutions to (2.20) are λ = ηj , for 0 ≤ j ≤ n − 1, where η = e2πi/n . (2.21) For each choice of j , we can try to find eigenvectors corresponding to η j . If we set x1 = 1, we can conclude from (2.19) that x2 = η j , x3 = η 2j , ... , xn = η (n−1)j , thereby obtaining a nonzero solution to the eigenvector equation. Thus for each j , 0 ≤ j ≤ n − 1, we do indeed have an eigenvector 1 ηj ej = η 2 j , · η (n−1)j for the eigenvalu...
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This document was uploaded on 01/12/2014.

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