2.1.11.2 Book 2

2 Background 1.2 Cosmological 1.2.11 Mathematics

Book 4 2.1.11.4

Book 3 - Determinants and Matrices

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

Contents

I. Preliminaries

01. Sum/product symbols
02. Mathematical induction
03. Polynomials
04. Permutations

II. Determinants

05. Determinants of second and third order
06. Determinants of order n

III. The most important properties of Determinants

07. Determinants, most important properties
08. Description of determinants according to Weierstrass
09. Determinant of transposed matrix
10. Expansion formulae
11. Evaluation of determinants
12. Cramer's Rule
13. Multiplication of determinants
IV. Matrices
14. Multiplication of matrices
15. Inverse Matrix
16. Group of regular matrices
17. Addition of matrices
18. Contragredient and orthogonal matrices

V. Vector spaces. Rank of a matrix

19. Vector spaces
20. Linear dependence
21. Relationship between different bases
22. Dimensions of partial spaces
23. Rank of matrix
24. Rank of a product
25. Orthogonal bases

VI. Linear Spaces

26. Homogeneous/ non-homogeneous equations
27. General solution of homogeneous system
28. Solubility of non-homogeneous systems of equations
29. Homogeneous variable
30. Numerical solution of linear equations

VII. Hermitian/Quadratic forms

31. Transformation of Hermitian forms
32. Characteristic roots. Eigen vectors.
33. Principal axes transformation
34. Definite Hermitian form

VIII. More about determinants and matrices

35. Vandermonde determinants
36. Hadamard's determinant estimate
37. Laplace's expansion rule
38. Partial matrices
39. Characteristic roots
40. Kronecker product
41. Cayley-Hamilton relations

X. Similarity

42. Classes ofIsimilarity
43. Linear mapping
44. Decomposition into components with a single characteristic root.
45. Decomposition into elementary components
46. Jordan's normal form
47. Similarity to diagonal matrices

Index

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