Vector spaces and matrices in physics /
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| Author / Creator: | Jain, M. C. |
|---|---|
| Imprint: | Boca Raton : CRC Press ; New Delhi : Narosa Pub. House, c2001. |
| Description: | xi, 183 p. : ill. ; 25 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4372802 |
Table of Contents:
- Preface
- 1.. Introduction
- 1.1. Abstract Algebraic Systems
- 1.2. Properties of Binary Operations
- 1.3. Group
- 1.4. Field
- 1.5. Ring
- 1.6. Functions or Mappings
- Exercises
- 2.. Vector Spaces
- 2.1. Generalization from Physical to Abstract Vectors
- 2.2. Vector Space
- 2.3. Subspace
- 2.4. Linear Combination of Vectors
- 2.5. Linear Independence and Dependence
- 2.6. Basis and Dimension: Coordinates
- 2.7. Isomorphism of Vector Spaces
- 2.8. Inner Product of Vectors
- 2.9. Norm (or length) of a Vector
- 2.10. Distance Between two Vectors
- 2.11. Schwarz Inequality
- 2.12. Orthogonality
- Exercises
- 3.. Linear Transformations
- 3.1. Definition
- 3.2. Equality
- 3.3. Sum and Scalar Multiple
- 3.4. Zero Transformation
- 3.5. Identity Transformation
- 3.6. Idempotent Transformation
- 3.7. Nilpotent Transformation
- 3.8. Nonsingular Transformation
- 3.9. Orthogonal Transformation
- Exercises
- 4.. Basic Matrix Algebra and Special Matrices
- 4.1. Definition
- 4.2. Equality
- 4.3. Sum and Difference of Matrices
- 4.4. Scalar Multiple of a Martix
- 4.5. Matrix Multiplication
- 4.6. Row and Column Vectors
- 4.7. Transpose of a Matrix
- 4.8. Conjugate of a Matrix
- 4.9. Conjugate-Transpose (Hermitian-Conjugate of a Matrix)
- 4.10. Trace of a Square Matrix
- 4.11. Determinant of a Square Matrix
- 4.12. Special Square Matrices
- 4.13. Adjoint of a Matrix
- 4.14. Determination of Inverse of a Matrix
- 4.15. Vector Space of Matrices
- Exercises
- 5.. Rank of a Matrix
- 5.1. Row and column Vectors of a Matrix
- 5.2. Rank of a Matrix
- 5.3. Row Space and Column Space of a Matrix
- 5.4. Elementary Row Operations on a Matrix
- Exercises
- 6.. Systems of Linear Equations
- 6.1. Homogeneous and Nonhomogeneous Linear Systems
- 6.2. Matrix Form of a Linear System
- 6.3. Existence and Uniqueness Theorems
- 6.4. A Practical Method of Solving Linear Systems: Gauss Elimination
- Exercises
- 7.. Matrices and Linear Transformations
- 7.1. Matrix Representation of a Linear Transformation
- 7.2. Representation of Product of Transformations
- 7.3. Change of Bases and Similarity Transformation
- Exercises
- 8.. Eigenvalues and Eigenvectors of a Matrix
- 8.1. Eigenvalues and Eigenvectors
- 8.2. Determination of Eigenvalues and Eigenvectors
- 8.3. Linear Independence of Eigenvectors
- 8.4. Eigenvalues and Eigenvectors of Similar Matrices
- 8.5. Eigenvalues of a Diagonal Matrix
- 8.6. Hermitian, Skew-Hermitian and Unitary Matrices
- 8.7. Diagonalization of a Matrix
- 8.8. Simultaneous Diagonalization and Commutativity
- 8.9. An Application: Reduction of Coupled Differential Equations to Matrix Eigenvalue Problem
- Exercises
- 9.. Caley-Hamilton Theorem. Minimal Polynomial of a Matrix
- 9.1. Caley-Hamilton Theorem
- 9.2. Determination of Inverse of a Matrix
- 9.3. Minimal Polynomial of a Matrix
- 9.4. A Criterion for Diagonalizability
- Exercises
- 10.. Functions of a Matrix
- 10.1. Power of a Matrix
- 10.2. Matrix Polynomial
- 10.3. Matrix Power Series
- 10.4. Evaluation of Matrix Functions for Diagonalizable Matrices
- 10.5. Evaluation of Matrix Functions Using Minimal Polynomial
- Exercises
- 11.. Bilinear, Quadratic, Hermitian and Skew-Hermitian Forms
- 11.1. Bilinear Form
- 11.2. Quadratic Form
- 11.3. Hermitian (Skew-Hermitian) Form
- 11.4. Reduction of a Quadratic Form to Canonial Form
- 11.5. Principal Axes Transformation
- Exercises
- Answers/Hints to Exercises
- Index
