| Preface |
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xi | |
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1 | (1) |
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2 | (1) |
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Definition of a Determinant |
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3 | (3) |
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Elementary Properties of Determinants |
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6 | (3) |
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9 | (5) |
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Cramer's Rule for Linear Equations |
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14 | (2) |
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Minors and Rank of Matrices |
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16 | (9) |
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18 | (4) |
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22 | (3) |
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25 | (1) |
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Addition and Multiplication |
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26 | (2) |
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28 | (5) |
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33 | (2) |
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35 | (3) |
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38 | (9) |
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43 | (3) |
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46 | (1) |
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Solution of Linear and Nonlinear Systems |
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47 | (1) |
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48 | (7) |
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Gauss Elimination with Pivoting |
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55 | (3) |
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Computing the Inverse of a Matrix |
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58 | (3) |
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61 | (5) |
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66 | (12) |
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Iterative Methods for Solving Ax = b |
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78 | (7) |
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85 | (38) |
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108 | (13) |
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121 | (2) |
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General Theory of Solvability of Linear Algebraic Equations |
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123 | (1) |
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Sylvester's Theorem and the Determinants of Matrix Products |
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124 | (5) |
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Gauss-Jordan Transformation of a Matrix |
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129 | (4) |
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General Solvability Theorem for Ax = b |
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133 | (17) |
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Linear Dependence of a Vector Set and the Rank of Its Matrix |
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150 | (5) |
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The Fredholm Alternative Theorem |
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155 | (8) |
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159 | (2) |
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161 | (2) |
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163 | (2) |
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Linear Operators in a Normed Linear Vector Space |
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165 | (5) |
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Basis Sets in a Normed Linear Vector Space |
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170 | (9) |
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179 | (5) |
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Some Special Properties of Eigenvalues |
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184 | (5) |
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Calculation of Eigenvalues |
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189 | (16) |
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196 | (7) |
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203 | (2) |
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205 | (1) |
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Implications of the Spectral Resolution Theorem |
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206 | (7) |
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Diagonalization by a Similarity Transformation |
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213 | (6) |
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Matrices with Distrinct Eigenvalues |
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219 | (1) |
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Unitary and Orthogonal Matrices |
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220 | (5) |
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Semidiagonalization Theorem |
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225 | (2) |
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227 | (18) |
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245 | (4) |
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249 | (5) |
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The Initial Value Problem |
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254 | (5) |
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259 | (20) |
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261 | (17) |
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278 | (1) |
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Imperfect or Defective Matrices |
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279 | (1) |
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Rank of the Characteristic Matrix |
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280 | (2) |
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Jordan Block Diagonal Matrices |
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282 | (6) |
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The Jordan Canonical Form |
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288 | (6) |
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Determination of Generalized Eigenvectors |
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294 | (9) |
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Dyadic Form of an Imperfect Matrix |
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303 | (1) |
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Schmidt's Normal Form of an Arbitrary Square Matrix |
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304 | (4) |
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The Initial Value Problem |
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308 | (7) |
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310 | (4) |
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314 | (1) |
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Infinite-Dimensional Linear Vector Spaces |
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315 | (1) |
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Infinite-Dimensional Spaces |
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316 | (3) |
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Riemann and Lebesgue Integration |
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319 | (3) |
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322 | (2) |
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324 | (2) |
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326 | (4) |
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330 | (6) |
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Solutions to Problems Involving k-term Dyadics |
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336 | (7) |
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343 | (12) |
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351 | (2) |
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353 | (2) |
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Linear Integral Operators in a Hilbert Space |
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355 | (1) |
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356 | (10) |
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Completely Continuous and Hilbert-Schmidt Operators |
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366 | (9) |
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375 | (12) |
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Spectral Theory of Integral Operators |
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387 | (26) |
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406 | (5) |
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411 | (2) |
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Linear Differential Operators in a Hilbert Space |
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413 | (3) |
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The Differential Operator |
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416 | (4) |
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The Adjoint of a Differential Operator |
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420 | (6) |
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Solution to the General Inhomogeneous Problem |
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426 | (13) |
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Green's Function: Inverse of a Differential Operator |
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439 | (13) |
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Spectral Theory of Differential Operators |
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452 | (7) |
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Spectral Theory of Regular Sturm-Liouville Operators |
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459 | (18) |
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Spectral Theory of Singular Sturm-Liouville Operators |
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477 | (16) |
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Partial Differential Equations |
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493 | (18) |
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502 | (7) |
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509 | (2) |
| APPENDIX |
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A.1. Section 3.2: Gauss Elimination and the Solution to the Linear System Ax = b |
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511 | (3) |
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A.2. Example 3.6.1: Mass Separation with a Staged Absorber |
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514 | (1) |
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A.3. Section 3.7: Iterative Methods for Solving the Linear System Ax = b |
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515 | (3) |
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A.4. Exercise 3.7.2: Iterative Solution to Ax = b---Conjugate Gradient Method |
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518 | (1) |
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A.5. Example 3.8.1: Convergence of the Picard and Newton-Raphson Methods |
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519 | (2) |
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A.6. Example 3.8.2: Steady-State Solutions for a Continuously Stirred Tank Reactor |
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521 | (2) |
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A.7. Example 3.8.3: The Density Profile in a Liquid--Vapor Interface (Iterative Solution of an Integral Equation) |
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523 | (3) |
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A.8. Example 3.8.4: Phase Diagram of a Polymer Solution |
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526 | (3) |
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A.9. Section 4.3: Gauss--Jordan Elimination and the Solution to the Linear System Ax = b |
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529 | (2) |
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A.10. Section 5.4: Characteristic Polynomials and the Traces of a Square Matrix |
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531 | (2) |
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A.11. Section 5.6: Iterative Method for Calculating the Eigenvalues of Tridiagonal Matrices |
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533 | (1) |
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A.12. Example 5.6.1: Power Method for Iterative Calculation of Eigenvalues |
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534 | (1) |
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A.13. Example 6.2.1: Implementation of the Spectral Resolution Theorem---Matrix Functions |
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535 | (2) |
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A.14. Example 9.4.2: Numerical Solution of a Volterra Equation (Saturation in Porous Media) |
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537 | (3) |
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A.15. Example 10.5.3: Numerical Green's Function Solution to a Second-Order Inhomogeneous Equation |
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540 | (2) |
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A.16. Example 10.8.2: Series Solution to the Spherical Diffusion Equation (Carbon in a Cannonball) |
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542 | (1) |
| Index |
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543 | |
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