Book:A.C. Aitken/Determinants and Matrices/Eighth Edition

Subject Matter

 * Linear Algebra

Contents

 * Chapter $\text I$: Definitions and Fundamental Operations of Matrices
 * 1. Introductory
 * 2. Linear Equations and Transformations
 * 3. The Notation of Matrices
 * 4. Matrices, Row and Column Vectors, Scalars
 * 5. The Operations of Matrix Algebra
 * 6. Matrix Pre- and Postmultiplication
 * 7. Product of Three or More Matrices
 * 8. Transposition of Rows and Columns
 * 9. Transpose of a Product: Reversal Rule
 * 10. Algebraic Expressions in Matrix Notation
 * 11. Partitioned Matrices and Multiplication


 * Chapter $\text {II}$: Definition and Properties of Determinants
 * 12. The Solution of Simultaneous Equations
 * 13. Salient Properties of Eliminants
 * 14. Inversions and Class of Permutations
 * 15. Definition and Notation of Determinant
 * 16. Identity of Class of Conjugate Permutations
 * 17. Elementary Properties of Determinants
 * 18. Primeness of a Determinant
 * 19. Various Expansions of a Determinant
 * 20. Pivotal Evaluation of Determinants


 * Chapter $\text {III}$: Adjugate and Reciprocal Matrix: Solution of Simultaneous Equations: Rank and Linear Dependence
 * 21. The Adjugate Matrix of a Square Matrix
 * 22. Solution of Equations in Nonsingular Case
 * 23. Reversal Rule for Reciprocal of Product
 * 24. Orthogonal and Unitary Matrices
 * 25. Solution of Homogeneous Equations
 * 26. Rank and Nullity of Matrix
 * 27. Linear Dependence
 * 28. Conditions for Solution of Homogeneous Equations
 * 29. Reduction of Matrix to Equivalent Form
 * 30. Solution of Non-Honogeneous Equations


 * Chapter $\text {IV}$: Cauchy and Laplace Expansions: Multiplication Theorems
 * 31. Expansion by Elements of Row and Column
 * 32. Complementary Minors and Cofactors
 * 33. Laplacian Expansion of a Determinant
 * 34. Multiplication of Determinants
 * 35. Extended Laplace and Cauchy Expansions
 * 36. Determinant of Product of Rectangular Matrices
 * 37. Expansion by Diagonal Elements: Normal Form


 * Chapter $\text V$: Compound Matrices and Determinants: Dual Theorems
 * 38. Compound and Adjugate Compound Matrices
 * 39. Binet-Cauchy Theorem on Product of Compounds
 * 40. Reciprocal of Nonsingular Compound Matrix
 * 41. Rank Expressed by Compound Matrices
 * 42. Jacobi's Theorem on Minors of Adjugate
 * 43. Franke's Theorem on Minors of Compound
 * 44. Hybrid Compounds of Bazin and Reiss
 * 45. Complementary and Extensional Identities
 * 46. Schweinsian Expansions of Determinant Quotients


 * Chapter $\text {VI}$: Special Determinants: Alternant, Persymmetric, Bigradient, Centrosymmetric, Jacobian, Hessian, Wronskian
 * 47. Alternant Matrices and Determinants
 * 48. Elementary and Complete Symmetric Functions
 * 49. Bialternant Symmetric Functions of Jacobi
 * 50. Confluent or Differentiated Alternants
 * 51. Persymmetric, Circulant and Centrosymmetric
 * 52. Dialytic Elimination: Bigradients
 * 53. Continuant Matrices and Continuants
 * 54. Jacobians, Hessians and Wronskians
 * Additional Examples
 * Index



Source work progress
* : Chapter $\text I$: Definitions and Fundamental Operations of Matrices: $5$. The Operations of Matrix Algebra