Introduction to the theory of canonical matrices 规范型矩阵导论

Thorough and self-contained, this penetrating study of the theory of canonical matrices presents a detailed consideration of all the theory’s principal features. Topics include elementary transformations and bilinear and quadratic forms; canonical reduction of equivalent matrices; subgroups of the group of equivalent transformations; and rational and classical canonical forms. The final chapters explore several methods of canonical reduction, including those of unitary and orthogonal transformations. 1952 ed. Index. Appendix. Historical notes. Bibliographies. 275 problems.

CHAPTER Ⅰ DEFINITIONS AND FUNDAMENTAL PROPERTIES OF MATRICES
1. Introductory
2. Definitions and Fundamental Properties
3. Matrix Multip]ication
4. Reciprocal of a Non-Singular Matrix
5. The Reversal Law in Transposed and Reciprocal Products
6. Matrices Partitioned into Submatrices
7. Isolated Elements and Minors
8. Historical Note
CHAPTER Ⅱ ELEMENTARY TRANSFORMATIONS. BILINEAR AND QUADRATIC FORMS
1. The Solution of n Linear Equations in n Unknowns
2. Interchange of Rows and Columns in a Determinant or Matrix
3. Linear Combination of Rows or Columns in a Determinant or Matrix
4. Multiplication of Rows or Columns
5. Linear Transformation of Variables
6. Bilinear and Quadratic Forms
7.The Highest Common Factor of Two Polynomials
8.Historical Note
CHAPTER Ⅲ THE CANONICAL REDUCTION OF EQUIVALENT MATRICES
1. General Linear Transformation
2. Equivalent Matrices in a Field
3. The Equivalence of Matrices with Integer Elements
4. Polynomials with Matrix Coefficients: ),-Matrices
5. The H.C.F. Process for Polynomials
6. Smith's Canonical Form for Equivalent Matrices
7. The H.C.F. of m-rowed Minors of a ),-Matrix
8. Equivalent ),-Matrices
9. Observations on the Theorems
10. The Singular Case of n Linear Equations in n Variables
11. Historical Note
CHAPTER Ⅳ SUBGROUPS OF THE GROUP OF EQUIVALENT TRANSFORMATIONS
1. Matrices of Special Type, Symmetric, Orthogonal, &c.
2.Axisymmetric, Hermitian, Orthogonal, and Unitary Matrices
3.Special Subgroups of the Group of Equivalent Transformations
4. Quadratic and Bilinear Forms associated with the Subgroups
5. Geometrical Interpretation of the Collineation
6. The Poles and Latent Points of a Collineation
7. Change of Frame of Reference
8. Alternative Geometrical Interpretation
9. The Cayley-Hamilton Theorem
10. Historical Note
CHAPTER Ⅴ A RATIONAL CANONICAL FORM FOR THE COLLINEATORY GROUP
1. Linear Independence of Vectors in a Field
2. The Reduced Characteristic Function of a Vector
3. Fundamental Theorem of the Reduced Characteristic Function
4. A Rational Canonical Form for Collineatory Transformations
5. Properties of the R.C.F.'s of the Canonical Vectors
6. Observations upon the Theorems
7. Geometrical and Dual Aspect of Theorem II
8. The Invariant Factors of the Characteristic Matrix of B
9. Historical Note
CHAPTER Ⅵ THE CLASSICAL CANONTD
CHAPTER Ⅶ CONGRUENT AND CONJUNCTIVE TRANSFORMATIONS:QUADRATIC AND HERMITIAN FORMS
CHAPTER Ⅷ CANONICAL REDUCTION BY UNITARY AND ORTHOGONAL TRANSFORMATIONS
CHAPTER Ⅸ TEH CANONICAL REDUCTION OF PENCILS OF MATRICES
CHAPTER Ⅹ APPLICATIONS OF CANONICAL FORMS TO SOLUTION OF LINEAR MATRIX EQUATIONS.COMMUTANTS AND INVARIANTS
CHAPTER Ⅺ PRACTICAL APPLICATIONS OF CANONICAL REDUCTION
APPENDIX
MISCELLANEOUS EXAMPLES
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