Matrices - Matrices Kevin W. Cassel Mechanical, Materials...

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Unformatted text preview: Matrices Kevin W. Cassel Mechanical, Materials and Aerospace Engineering Department Illinois Institute of Technology 10 West 32nd Street Chicago, IL 60616 cassel@iit.edu c 2010 Kevin W. Cassel Contents 1 Vectors and Matrices 3 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . 11 1.5 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Cramers Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7 Linear Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7.1 Vectors in 3-D - Review . . . . . . . . . . . . . . . . . . . 17 1.7.2 n-Dimensional Vectors . . . . . . . . . . . . . . . . . . . . 18 1.7.3 Linear Dependence of Vectors . . . . . . . . . . . . . . . . 19 1.7.4 Basis of a Vector Space . . . . . . . . . . . . . . . . . . . 21 1.7.5 Gram-Schmidt Orthogonalization . . . . . . . . . . . . . . 23 1.8 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . 25 2 The Eigenproblem and Its Applications 28 2.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 28 2.2 Real Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . 34 2.2.1 Linear Systems of Equations . . . . . . . . . . . . . . . . 35 2.2.2 Iterative Numerical Methods . . . . . . . . . . . . . . . . 37 2.2.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Nonsymmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . 44 2.4 Systems of Ordinary Differential Equations . . . . . . . . . . . . 47 2.4.1 General Approach . . . . . . . . . . . . . . . . . . . . . . 47 2.4.2 Applications to Dynamical Systems . . . . . . . . . . . . 56 2.4.3 Stability of Dynamical Systems . . . . . . . . . . . . . . . 61 2.5 Decomposition of Matrices . . . . . . . . . . . . . . . . . . . . . . 72 2.5.1 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . 72 2.5.2 Singular Value Decomposition . . . . . . . . . . . . . . . . 73 2.6 Functions of Matrices . . . . . . . . . . . . . . . . . . . . . . . . 76 1 CONTENTS 2 3 Eigenfunction Expansions of Diff. Eqns. 80 3.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2 Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . 85 3.3 Self-Adjoint Differential Operators . . . . . . . . . . . . . . . . . 92 3.4 Sturm-Liouville Equation . . . . . . . . . . . . . . . . . . . . . . 95 3.4.1 Requirement for Self-Adjoint Operator . . . . . . . . . . . 95 3.4.2 Eigenfunctions of Sturm-Liouville Operators . . . . . . . . 96 3.5 Column Buckling Example . . . . . . . . . . . . . . . . . . . . . . 101 3.6 Eigenfunction Solutions of Partial Differential Equations . . . . . 105 3.6.1 Laplaces Equation . . . . . . . . . . . . . . . . . . . . . . 105 3.6.2 Application to Vibrations of Continuous Systems . . . . . 110Application to Vibrations of Continuous Systems ....
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Matrices - Matrices Kevin W. Cassel Mechanical, Materials...

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