Topics In Linear Algebra and Its Applications - STEM2

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M1M2K1 K2 K3Figure 1: Coupled springs, With masses m 1 and m 1 . The spring constantsare k 1 , k 2 and k 3 . k 2 is the constant for the coupling spring and is muchsmaller than the other two constants.32
withandA = m 2B = −2(k + k 2 )mC = (k + k 2 ) 2 − k 2 2 = k2 + 2kk 2 .Let a = (k + k 2 − λm) and b = −k 2 . Then since the determinant is 0, wemust havea 2 = b 2So eitherora = b,a = −b.The equations for the eigenvectors areandIf a = b, then the eigenvector isaU 1 + bU 2 = 0,bU 1 + aU 2 = 0.[1−1]On the other hand, if a = −b, then the eigenvector is[11]The eigenvalues are λ 1 and λ 2 . Let a i be the value of a for eigenvalue λ i , andlet b i be the value of b for eigenvalue λ i . Now let us compute with values assignedto the constants. Let m = 1, k = 1, and k 2 = 1/10. The computationis done with a Matlab script (Octave script). Call it coupled.m:33
Page 6 and 7: The scalar coefficients are called
Page 9 and 10: 2. Adding a multiple of one column
Page 11 and 12: Let u 1 , u 2 , ..., u p be a basis
Page 13 and 14: 11 Gram-Schmidt OrthogonalizationGi
Page 15 and 16: Conversely, if we have equality, th
Page 17 and 18: So Bessel’s inequality is an equa
Page 19 and 20: ⎡[ ]⎢ x1 x 2 x 3 ⎣1 5 25 25 1
Page 21 and 22: 19 Rotation MatricesSee the paper R
Page 23 and 24: 22 The Characteristic Polynomial an
Page 25 and 26: 26 The Spectral TheoremVarious vers
Page 27 and 28: 28.2 Decoupling EquationsConsider t
Page 29 and 30: We haveOn the other handX T i KX j
Page 31: In matrix form the equations become
Page 35 and 36: a 1 and b 1 are equal, so the eigen
Page 37 and 38: Figure 2: Coupled oscillations. The
Page 39 and 40: The modulating frequency isφ = 0.0
Page 41 and 42: 29 Polynomial Roots, The Frobenius
Page 43 and 44: where n and m have no common factor
Page 45 and 46: UsingandE =p =mc 2√1 − v 2 /c 2
Page 47 and 48: 40 BibliographyThe number of books

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