If x is an eigenpair then xt axxt x bx 2 x 2 0 proof

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Unformatted text preview: the mathematical model would be grossly inconsistent with reality if the symmetric matrix A in (7.6.2) were not positive definite. It turns out that A is positive definite because there is a Cholesky factorization A = RT R with ⎞ ⎛ r −1/r 1 R= ⎜ T⎜ ⎜ mL ⎜ ⎝ 1 r2 −1/r2 .. . rn−1 .. . −1/rn−1 rn ⎟ ⎟ ⎟ ⎟ ⎠ with rk = 2− k−1 , k and thus we are insured that each λk > 0. In fact, since A is a tridiagonal Toeplitz matrix, the results of Example 7.2.5 (p. 514) can be used to show that λk = Copyright c 2000 SIAM 2T mL 1 − cos kπ n+1 = kπ 4T sin2 mL 2(n + 1) (see Exercise 7.2.18). Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 562 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Therefore, √ √ ⎧ ⎫ = αk cos t λk + βk sin t λk ⎬ ⎨ zk √ zk (0) = ck ˜ ˜ =⇒ zk = ck cos t λk , (7.6.3) ⎩ ⎭ zk (0) = 0 and for P = x1 | x2 | · · · | xn , n It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] y = Pz = z1 x1 + z2 x2 + · · · + zn xn = √ cj cos t λk ˜ xj . (7.6.4) j =1 D E This means that every possible mode of vibration is a combination of modes determined by the eigenvectors xj . To understand this more clearly, suppose that the beads are initially positioned according to the components of xj —i.e., ˜ c = y(0) = xj . Then c = PT c = PT xj = ej , so (7.6.3) and (7.6.4) reduce to √ if k = j =⇒ y = cos t√λ zk = cos t λk (7.6.5) xj . k 0 if k = j th T H In other words, when y(0) = xj , the j eigenpair (λj , xj ) completely determines the mode of vibration because the amplitudes are determined by xj , and each bead vibrates with a common frequency f = λj /2π. This type of motion (7.6.5) is called a normal mode of vibration. In these terms, equation (7.6.4) translates to say that every possible mode of vibration is a combination of the normal modes . For example, when n = 3, the matrix in (7.6.2) is ⎧ ⎫ ⎛ ⎞ 2 −1 0 ⎨ λ1 = (T /mL)(2) ...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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