Athanasios Dermanis

LINEAR ALGEBRA AND MATRIX THEORY

Editions Ziti, Thessaloniki 1985 (in Greek)


 
CONTENTS
Preface

Bibliography
1. Matrices

1. Definition and general characteristics of matrices - 2. Matrix operations - 3. Special matrix types - 4. Properties of matrix operations - Exercises

2. Transpose of a matrix
1. Definition - 2. Properties - 3. Symmetric and antisymmetric matrices - Exercises
3. Determinants
1. General characteristics - 2. Development of a determinant by rows and by columns - 3. Properties of determinants - Exercises
4. Inverse of a matrix
1. General characteristics - 2. Properties of inverses - 3. Computation of inverses - 4. Orthogonal matrices - Exercises
5. Rank of a matrix
1. Linear independence - 2. Rank of a matrix - 3. Rank properties - Exercises
6. Composite matrices

1. General characteristics - 2. Rank and determinant of a composite matrix - 3. Inverse of a composite matrix - 4. Special types of matrix products - Exercises

7. Relations between matrices

1. Elementary matrix transformations - 2. Reduction of a matrix to scalar form - 3. Reduction of a matrix to normal form - 4. Matrix equivalence - 5. Matrix similarity - 6. Matrix congruence - 7. Orthogonal matrix congruence - 8. Full rank factorization - Exercises

8. Eigenvalues and eigenvectors
1. General characteristics - 2. Properies of eigenvalues - 3. Properies of eigenvectors - 4. Eigenvalues and eigenvectors of symmetric matrices - Exercises
9. Bilinear and quadratic forms
1. Bilinear forms - 2. Quadratic forms - 3. Reduction of a quadratic form to diagonal form - 4. Reduction of a quadratic form to normal form - 5. Definite, semi-definite and indefinite matrices - 6. Properties of quadratic forms - Exercises
10. Complex matrices
Exercises
11. Matrix differentiation
1. Differentiation and integration of matrices - 2. Linearization - Exercises
12. Systems of linear equations
1. The role of linear equation systems in applied sciences - 2. General characteristics of systems of linear equations - 3. Compatibility of linear equations - 4. The homogeneous system of linear equations - 5. The number of solutions for a system in n equations and m unknowns - 6. Compatibility relations and the general solution of a system of linear equations - 7. Least squares solution of systems of linear equations - Exercises
13. Generalized inverses
1. Left and right inverses - 2. Pseudoinverse - 3. Generalized inverse, least squares and minimum norm generalized inverse - Exercises
14. Plane vectors
1. Analytical geometry of the plane - 2. Planar vectors - 3. Length and inner product of vectors - 4. Change of base vectors
15. Linear mappings on a plane
1. Linear mappings - 2. Properties of linear mappings of the plane - 3. Diagonalization of a linear mapping - 4. Mapping of the plane into a line and of a line into a plane
16. Vectors in three-dimensional Euclidean space
1. Analytical geometry of three dimensional Euclidean space - 2. Vectors in three dimensional Euclidean space - 3. Length, inner and outer product
17. Linear mappings in three dimensions
1. Linear mappings from one three dimensional space into another - 2. Linear mappings of three dimensional space without full rank - 3. Linear mappings from one three dimensional space into a plane or line - 4. Linear mappings from a plane or line into a three dimensional space
18. Linear spaces
1. General characteristics - 2. Linear independence, basis and dimension - 3. Distance, norm and inner product
19. Linear mappings
1. Linear mappings from one linear space into another - 2. Linear operators - 3. The linear equation A x = y
20. Introduction to tensors
1. The role of tensors in physics - 2. Covariant and contravariant components of vectors in three dimensional Euclidean spaces - 3. Tensors - 4. Addition, tensor producr, contraction and inner product of tensors - 5. Curvilinear coordinates and tensor fields - 6. Element of length and the metric tensor - 7. Covariant differentiation
Index