Amazoncomexecobidosasin0898714540 577 it is illegal to

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Unformatted text preview: apply the technique used in the vibrating beads problem in Example 7.6.1 (p. 559) to determine the normal modes. Compare the results with those of part (b). D E T H IG R 7.6.3. Three masses m1 , m2 , and m3 are suspended on three identical springs (with spring constant k ) as shown below. Each mass is initially displaced from its equilibrium position by a vertical distance and then released to vibrate freely. (a) If yi (t) denotes the displacement of mi from equilibrium at time t, show that the mass-stiffness equation is My = Ky, where ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ m1 0 y1 (t) 0 2 −1 0 M = ⎝ 0 m2 0 ⎠, y = ⎝ y2 (t) ⎠ K = k ⎝ −1 , 2 −1 ⎠ 0 0 m3 y3 (t) 0 −1 1 Y P O C ( k33 = 1 is not a mistake!). (b) Show that K is positive definite. (c) Find the normal modes when m1 = m2 = m3 = m. 7.6.4. By diagonalizing the quadratic form 13x2 + 10xy + 13y 2 , show that the rotated graph of 13x2 + 10xy + 13y 2 = 72 is an ellipse in standard form as shown in Figure 7.2.1 on p. 505. 7.6.5. Suppose that A is a real-symmetric matrix. Explain why the signs of the pivots in the LDU factorization for A reveal the inertia of A. Copyright c 2000 SIAM Buy online from SIAM Buy from 572 Chapter 7 Eigenvalues and Eigenvectors 7.6.6. Consider the quadratic form It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] f (x) = 1 (−2x2 + 7x2 + 4x2 + 4x1 x2 + 16x1 x3 + 20x2 x3 ). 1 2 3 9 (a) Find a symmetric matrix A so that f (x) = xT Ax. (b) Diagonalize the quadratic form using the LDLT factorization as described in Example 7.6.4, and determine the inertia of A. (c) Is this a positive definite form? (d) Verify the inertia obtained above is correct by computing the eigenvalues of A. (e) Verify Sylvester’s law of inertia by making up a congruence transformation C and then computing the inertia of CT AC. D E T H 7.6.7. Polar Factorization. Explain why each nonsingular A ∈ C n×n c...
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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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