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, semidefinite 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 threedimensional 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 