Determinants and Matrices

from Dr. R. Kochendörfer's "Determinants and Matrices", published by Teubner in Leipzig in 1961.


I. Preliminaries   24. Rank of a product
1. Sum/product symbols   25. Orthogonal bases
2. Mathematical induction  

VI Linear Spaces

3. Polynomials   26. Homogeneous/ non-homogeneous equations
4. Permutations   27. General solution of homogeneous system

II. Determinants

  28. Solubility of non-homogeneous systems of equations
5. Determinants of second and third order   29. Homogeneous variable
6. Determinants of order n   30. Numerical solution of linear equations

III. The most important properties of Determinants


VII Hermitian/Quadratic forms

7. Determinants, most important properties   31. Transformation of Hermitian forms
8. Description of determinants according to Weierstrass   32. Characteristic roots. Eigen vectors.
9. Determinant of transposed matrix   33. Principal axes transformation
10. Expansion formulae   34. Definite Hermitian form
11. Evaluation of determinants  

VIII. More about determinants and matrices

12. Cramer's Rule   35. Vandermonde determinants
13. Multiplication of determinants   36. Hadamard's determinant estimate
IV. Matrices  

37. Laplace's expansion rule

14. Multiplication of matrices   38. Partial matrices
15. Inverse Matrix   39. Characteristic roots
16. Group of regular matrices   40. Kronecker product
17. Addition of matrices   41. Cayley-Hamilton relations
18. Contragredient and orthogonal matrices  

X. Similarity

V. Vector spaces. Rank of a matrix

  42. Classes of similarity
19. Vector spaces   43. Linear mapping
20. Linear dependence   44. Decomposition into components with a single characteristic root.
21. Relationship between different bases   45. Decomposition into elementary components
22. Dimensions of partial spaces   46. Jordan's normal form
23. Rank of matrix   47. Similarity to diagonal matrices