Topics In Linear Algebra and Its Applications - STEM2

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Kinetic Energy:Lagrangian:Lagrange’s Equation:We haveSo Lagrange’s equation isTrying e αt as a solution, we findT = mẋ22L = T − V.d ∂Ldt dẋ − ∂L∂x = 0.∂Ldẋ = mẋ.d ∂Ldt dẋ = mẍ.∂L∂x = −kxmẍ + kx = 0.(mα 2 + k)e αt = 0orwhere√α = i k/m = iω,ω 2 = k m .Sox = e iωtThe differential equation can be written as∂ 2∂t 2x = −ω2 x.So x = e iωt and x = e −iωt are eigenfunctions, and −ω 2 is an eigenvalue ofthe linear operator∂ 2∂t 2.26
28.2 Decoupling EquationsConsider the equationMẌ + KX = F,where M is a square n-dimensional symmetric mass matrix, K is a squaren-dimensional symmetric stiffness matrix, and F is a n-dimensional force.Both M and K must be positive definite because they represent the kineticand strain quadratic forms. If either of them were not positive definite, therewould be a nodal displacement or a nodal displacement velocity giving anegative energy. We assume a solution of the formX = X 0 exp(iωt).Consider the homogeneous problem with F = 0. We haveLet us write X for X 0 . Then we have−Mω 2 X 0 + KX 0 = 0.(K − ω 2 M)X = 0This is a generalized eigenvalue problem. This homogeneous problem has anonzero solution if and only if the determinant is zero:|K − ω 2 M| = 0.This determinant is called the characteristic function, which is an nth degreepolynomial in the variable ω 2 . The roots are called eigenvalues. In the caseof symmetric matrices the eigenvalues are real. To show this, let the innerproduct ben∑(u, v) = u i ¯v i ,where the subscripted values are the vector components, and where the barindicates the complex conjugate. Then if λ is an eigenvalue, with eigenvectorv, because M and K are self-adjoint (real symmetric), we havei=1λ(Mv, v) = (λMv, v) = (Kv, v) = (v, ¯K T v)= (v, Kv) = (v, λMv) = ¯λ(v, Mv) = ¯λ( ¯M T v, v) = ¯λ(Mv, v).27
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: 26 The Spectral TheoremVarious vers
Page 29 and 30: We haveOn the other handX T i KX j
Page 31 and 32: In matrix form the equations become
Page 33 and 34: withandA = m 2B = −2(k + k 2 )mC
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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