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  • Applications of Linear Algebra

    by

    Gordon C. Everstine

    5 June 2010

  • Copyright c 19982010 by Gordon C. Everstine.All rights reserved.

    This book was typeset with LATEX 2 (MiKTeX).

  • Preface

    These lecture notes are intended to supplement a one-semester graduate-level engineeringcourse at The George Washington University in algebraic methods appropriate to the solutionof engineering computational problems, including systems of linear equations, linear vectorspaces, matrices, least squares problems, Fourier series, and eigenvalue problems. In general,the mix of topics and level of presentation are aimed at upper-level undergraduates and first-year graduate students in mechanical, aerospace, and civil engineering.

    Gordon EverstineGaithersburg, MarylandJune 2010

    iii

  • Contents

    1 Systems of Linear Equations 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Nonsingular Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Diagonal and Triangular Systems . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Operation Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Partial Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 LU Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 Matrix Interpretation of Back Substitution . . . . . . . . . . . . . . . . . . . 141.9 LDU Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.10 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.11 Multiple Right-Hand Sides and Matrix Inverses . . . . . . . . . . . . . . . . 181.12 Tridiagonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.13 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    2 Vector Spaces 252.1 Rectangular Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Spanning a Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Row Space and Null Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Pseudoinverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Orthogonal Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8 Projections Onto Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    3 Change of Basis 403.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Examples of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    4 Least Squares Problems 484.1 Least Squares Fitting of Data . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 The General Linear Least Squares Problem . . . . . . . . . . . . . . . . . . . 524.3 Gram-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . 534.4 QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

    5 Fourier Series 555.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 Generalized Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Fourier Expansions Using a Polynomial Basis . . . . . . . . . . . . . . . . . 615.4 Similarity of Fourier Series With Least Squares Fitting . . . . . . . . . . . . 64

    v

  • 6 Eigenvalue Problems 646.1 Example 1: Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . . 656.2 Properties of the Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . 686.3 Example 2: Principal Axes of Stress . . . . . . . . . . . . . . . . . . . . . . . 736.4 Computing Eigenvalues by Power Iteration . . . . . . . . . . . . . . . . . . . 756.5 Inverse Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.6 Iteration for Other Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 796.7 Similarity Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.8 Positive Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.9 Application to Differential Equations . . . . . . . . . . . . . . . . . . . . . . 846.10 Application to Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . 88

    Bibliography 90

    Index 93

    List of Figures

    1 Two Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Some Vectors That Span the xy-Plane. . . . . . . . . . . . . . . . . . . . . . 303 90 Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Reflection in 45 Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Projection Onto Horizontal Axis. . . . . . . . . . . . . . . . . . . . . . . . . 346 Rotation by Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Projection Onto Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Element Coordinate Systems in the Finite Element Method. . . . . . . . . . 409 Basis Vectors in Polar Coordinate System. . . . . . . . . . . . . . . . . . . . 4110 Change of Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111 Element Coordinate System for Pin-Jointed Rod. . . . . . . . . . . . . . . . 4712 Example of Least Squares Fit of Data. . . . . . . . . . . . . . . . . . . . . . 5013 The First Two Vectors in Gram-Schmidt. . . . . . . . . . . . . . . . . . . . . 5414 The Projection of One Vector onto Another. . . . . . . . . . . . . . . . . . . 5815 Convergence of Series in Example. . . . . . . . . . . . . . . . . . . . . . . . . 5916 A Function with a Jump Discontinuity. . . . . . . . . . . . . . . . . . . . . . 6117 The First Two Vectors in Gram-Schmidt. . . . . . . . . . . . . . . . . . . . . 6318 2-DOF Mass-Spring System. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519 Mode Shapes for 2-DOF Mass-Spring System. (a) Undeformed shape, (b)

    Mode 1 (masses in-phase), (c) Mode 2 (masses out-of-phase). . . . . . . . . . 6720 3-DOF Mass-Spring System. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521 Geometrical Interpretation of Sweeping. . . . . . . . . . . . . . . . . . . . . 7922 Elastic System Acted Upon by Forces. . . . . . . . . . . . . . . . . . . . . . 83

    vi

  • 1 Systems of Linear Equations

    A system of n linear equations in n unknowns can be expressed in the form

    A11x1 + A12x2 + A13x3 + + A1nxn = b1A21x1 + A22x2 + A23x3 + + A2nxn = b2

    ...An1x1 + An2x2 + An3x3 + + Annxn = bn.

    (1.1)The problem is to find numbers x1, x2, , xn, given the coefficients Aij and the right-handside b1, b2, , bn.

    In matrix notation,Ax = b, (1.2)

    where A is the n n matrix

    A =

    A11 A12 A1nA21 A22 A2n

    ......

    . . ....

    An1 An2 Ann

    , (1.3)b is the right-hand side vector

    b =

    b1b2...bn

    , (1.4)and x is the solution vector

    x =

    x1x2...xn

    . (1.5)

    1.1 Definitions

    The identity matrix I is the square diagonal matrix with 1s on the diagonal:

    I =

    1 0 0 00 1 0 00 0 1 0...

    ......

    . . ....

    0 0 0 1

    . (1.6)

    In index notation, the identity matrix is the Kronecker delta:

    Iij = ij =

    {1, i = j,0, i 6= j. (1.7)

    1

  • The inverse A1 of a square matrix A is the matrix such that

    A1A = AA1 = I. (1.8)

    The inverse is unique if it exists. For example, let B be the inverse, so that AB = BA = I.If C is another inverse, C = CI = C(AB) = (CA)B = IB = B.

    A square matrix A is said to be orthogonal if

    AAT = ATA = I. (1.9)

    That is, an orthogonal matrix is one whose inverse is equal to the transpose.An upper triangular matrix has only zeros below the diagonal, e.g.,

    U =

    A11 A12 A13 A14 A150 A22 A23 A24 A250 0 A33 A34 A350 0 0 A44 A450 0 0 0 A55

    . (1.10)Similarly, a lower triangular matrix has only zeros above the diagonal, e.g.,

    L =

    A11 0 0 0 0A21 A22 0 0 0A31 A32 A33 0 0A41 A42 A43 A44 0A51 A52 A53 A54 A55

    . (1.11)A tridiagonal matrix has all its nonzeros on the main diagonal and the two adjacent diagonals,e.g.,

    A =

    x x 0 0 0 0x x x 0 0 00 x x x 0 00 0 x x x 00 0 0 x x x0 0 0 0 x x

    . (1.12)Consider two three-dimensional vectors u and v given by

    u = u1e1 + u2e2 + u3e3, v = v1e1 + v2e2 + v3e3, (1.13)

    where e1, e2, and e3 are the mutually orthogonal unit basis vectors in the Cartesian coordi-nate directions, and u1, u2, u3, v1, v2, and v3 are the vector components. The dot product(or inner product) of u and v is

    u v = (u1e1 + u2e2 + u3e3) (v1e1 + v2e2 + v3e3) = u1v1 + u2v2 + u3v3 =3i=1

    uivi, (1.14)

    whereei ej = ij. (1.15)

    2

  • -7@@@@@@@@R a

    b c = a bb sin

    -a b cos

    Figure 1: Two Vectors.

    In n dimensions,

    u v =ni=1

    uivi, (1.16)

    where n is the dimension of each vector (the number of components). There are severalnotations for this product, all of which are equivalent:

    inner product notation (u,v)

    vector notation u v

    index notationni=1

    uivi

    matrix notation uTv or vTu

    The scalar product can also be written in matrix notation as

    uTv =[u1 u2 un

    ]

    v1v2...vn

    . (1.17)For u a vector, the length of u is

    u = |u| =

    u u =

    ni=1

    uiui =u21 + u

    22 + + u2n. (1.18)

    A unit vector is a vect