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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
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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
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572
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
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
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H 7.6.7. Polar Factorization. Explain why each nonsingular A ∈ C n×n c...

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