| |
| |
| |
| Determinants | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The Definition of a Determinant | |
| |
| |
| |
| Properties of Determinants | |
| |
| |
| |
| Row and Column Expansions | |
| |
| |
| |
| Vectors and Matrices | |
| |
| |
| |
| The Inverse Matrix | |
| |
| |
| |
| The Determinant of a Matrix Product | |
| |
| |
| |
| The Derivative of a Determinant | |
| |
| |
| |
| The Theory of Linear Equations | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Linear Vector Spaces | |
| |
| |
| |
| Basis and Dimension | |
| |
| |
| |
| Solvability of Homogeneous Equations | |
| |
| |
| |
| Evaluation of Rank by Determinants | |
| |
| |
| |
| The General m x n Inhomogeneous System | |
| |
| |
| |
| Least-Squares Solution of Unsolvable Systems | |
| |
| |
| |
| Matrix Analysis of Differential Equations | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Systems of Linear Differential Equations | |
| |
| |
| |
| Reduction to the Homogeneous System | |
| |
| |
| |
| Solution by the Exponential Matrix | |
| |
| |
| |
| Solution by Eigenvalues and Eigenvectors | |
| |
| |
| |
| Eigenvalues, Eigenvectors, and Canonical Forms | |
| |
| |
| |
| Matrices with Distinct Eigenvalues | |
| |
| |
| |
| The Canonical Diagonal Form | |
| |
| |
| |
| The Trace and Other Invariants | |
| |
| |
| |
| Unitary Matrices | |
| |
| |
| |
| The Gram-Schmidt Orthogonalization Process | |
| |
| |
| |
| Principal Axes of Ellipsoids | |
| |
| |
| |
| Hermitian Matrices | |
| |
| |
| |
| Mass-spring Systems | |
| |
| |
| Positive Definiteness | |
| |
| |
| Simultaneous Diagonalization | |
| |
| |
| |
| Unitary Triangularization | |
| |
| |
| |
| Normal Matrices | |
| |
| |
| |
| The Jordan Canonical Form | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Principal Vectors | |
| |
| |
| |
| Proof of Jordan's Theorem | |
| |
| |
| |
| Variational Principles and Perturbation Theory | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The Rayleigh Principle | |
| |
| |
| |
| The Courant Minimax Theorem | |
| |
| |
| |
| The Inclusion Principle | |
| |
| |
| |
| A Determinant-criterion for Positive Definiteness | |
| |
| |
| |
| Determinants as Volumes | |
| |
| |
| Hadamard's Inequality | |
| |
| |
| |
| Weyl's Inequalities | |
| |
| |
| |
| Gershgorin's Theorem | |
| |
| |
| |
| Vector Norms and the Related Matrix Norms | |
| |
| |
| |
| The Condition-Number of a Matrix | |
| |
| |
| |
| Positive and Irreducible Matrices | |
| |
| |
| |
| Perturbations of the Spectrum | |
| |
| |
| |
| Continuous Dependence of Eigenvalues on Matrices | |
| |
| |
| |
| Numerical Methods | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The Method of Elimination | |
| |
| |
| |
| Factorization by Triangular Matrices | |
| |
| |
| |
| Direct Solution of Large systems of Linear Equations | |
| |
| |
| |
| Reduction of Rounding Error | |
| |
| |
| |
| The Gauss-Seidel and Other Iterative Methods | |
| |
| |
| |
| Computation of Eigenvectors from Known Eigenvalues | |
| |
| |
| |
| Numerical Instability of the Jordan Canonical Form | |
| |
| |
| |
| The Method of Iteration for Dominant Eigenvalues | |
| |
| |
| |
| Reduction to Obtain the Smaller Eigenvalues | |
| |
| |
| |
| Eigenvalues and Eigenvectors of Tridiagonal and Hessenberg Matrices | |
| |
| |
| |
| The Method of Householder and Bauer | |
| |
| |
| |
| Numerical Identification of Stable Matrices | |
| |
| |
| |
| Accurate Unitary Reduction to Triangular Form | |
| |
| |
| |
| The QR Method for Computing Eigenvalues | |
| |
| |
| Index | |