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Product Code:  CHEL/131.H 
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Hardcover ISBN:  9780821813768 
eBook: ISBN:  9781470469634 
Product Code:  CHEL/131.H.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $120.60 $91.35 
Hardcover ISBN:  9780821813768 
Product Code:  CHEL/131.H 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $62.10 
eBook ISBN:  9781470469634 
Product Code:  CHEL/131.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $58.50 
Hardcover ISBN:  9780821813768 
eBook ISBN:  9781470469634 
Product Code:  CHEL/131.H.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $120.60 $91.35 

Book DetailsAMS Chelsea PublishingVolume: 131; 1959; 384 ppMSC: Primary 15
This treatise, by one of Russia's leading mathematicians, gives in easily accessible form a coherent account of matrix theory with a view to applications in mathematics, theoretical physics, statistics, electrical engineering, etc. The individual chapters have been kept as far as possible independent of each other, so that the reader acquainted with the contents of Chapter 1 can proceed immediately to the chapters of special interest. Much of the material has been available until now only in the periodical literature.

Table of Contents

Front Cover

PREFACE

PUBLISHERS' PREFACE

CONTENTS

CHAPTER I: MATRICES AND OPERATIONS ON MATRICES

§ I. Matrices. Basic Notation

§ 2. Addition and Multiplication of Rectangular Matrices

§ 3. Square Matrices

§ 4. Compound Matrices. Minors of the Inverse Matrix

CHAPTER II: THE ALGORITHM OF GAUSS AND SOME OF ITS APPLICATIONS

§ 1. Gauss's Elimination Method

§ 2. Mechanical Interpretation of Gauss's Algorithm

§ 3. Sylvester's Determinant Identity

§ 4. The Decomposition of a Square Matrix into Triangular Factors

§ 5. The Partition of a Matrix into Blocks. The Technique of Operating with Partitioned Matrices. The Generalized Algorithm of Gauss

CHAPTER III: LINEAR OPERATORS IN AN nDIMENSIONAL VECTOR SPACE

§ I. Vector Spaces

§ 2. A Linear Operator Mapping an nDimensional Space into an mDimensional Space

§ 3. Addition and Multiplication of Linear Operators

§ 4. Transformation of Coordinates

§ 5. Equivalent Matrices. The Rank of an Operator. Sylvester's Inequality

§ 6. Linear Operators Mapping an nDimensional Space into Itself

§ 7. Characteristic Values and Characteristic Vectors of a Linear Operator

§ 8. Linear Operators of Simple Structure

CHAPTER IV: THE CHARACTERISTIC POLYNOMIAL AND THE MINIMAL POLYNOMIAL OF A MATRIX

§ I. Addition and Multiplication of Matrix Polynomials

§ 2. Right and Left Division of Matrix Polynomials

§ 3. The Generalized Bezout Theorem

§ 4. The Characteristic Polynomial of a Matrix. The Adjoint Matrix

§ 5. The Method of Faddeev for the Simultaneous Computation of the Coefficients of the Characteristic Polynomial and of the Adjoint Matrix

§ 6. The Minimal Polynomial of a Matrix

CHAPTER V: FUNCTIONS OF MATRICES

§ I. Definition of a Function of a Matrix

§ 2. The LagrangeSylvester Interpolation Polynomial

§ 3. Other Forms of the Definition of f(A). The Components of the Matrix A

§ 4. Representation of Functions of Matrices by means of Series

§ 5. Application of a Function of a Matrix to the Integration of a System of Linear Differential Equations with Constant Coefficients

§ 6. Stability of Motion in the Case of a Linear System

CHAPTER VI: EQUIVALENT TRANSFORMATIONS OF POLYNOMIAL MATRICES. ANALYTIC THEORY OF ELEMENTARY DIVISORS

§ 1. Elementary Transformations of a Polynomial Matrix

§ 2. Canonical Form of a λMatrix

§ 3. Invariant Polynomials and Elementary Divisors of a Polynomial Matrix

§ 4. Equivalence of Linear Binomials

§ 5. A Criterion for Similarity of Matrices

§ 6. The Normal Forms of a Matrix

§ 7. The Elementary Divisors of the Matrix f(A.)

§ 8. A General Method of Constructing the Transforming Matrix

§ 9. Another Method of Constructing a Transforming Matrix

CHAPTER VII: THE STRUCTURE OF A LINEAR OPERATOR IN AN nDIMENSIONAL SPACE (Geometrical Theory of Elementary Divisors)

§ 1. The Minimal Polynomial of a Vector and a Space (with Respect to a Given Linear Operator)

§ 2. Decomposition into Invariant Subspaces with CoPrime Minimal Polynomials

§ 3. Congruence. Factor Space

§ 4. Decomposition of a Space into Cyclic Invariant Subspaces

§ 5. The Normal Form of a Matrix

§ 6. Invariant Polynomials, Elementary Divisors

§ 7. The Jordan Normal Form of a Matrix

§ 8. K.rylov's Method of Transforming the Secular Equation

CHAPTER VIII: MATRIX EQUATIONS

§ 1. The Equation AX= XB

§ 2. The Special Case .4 = B. Commuting Matrices

§ 3. The Equation AX  XB = C

§ 4. The Scalar Equation f(X) = 0

§ 5. Matrix Polynomial Equations

§ 6. The Extraction of mth Roots of a NonSingular Matrix

§ 7. The Extraction of mth Roots of a Singular Matrix

§ 8. The Logarithm of a Matrix

CHAPTER IX: LINEAR OPERATORS IN A UNITARY SPACE

§ 1. General Considerations

§ 2. Metrization of a Space

§ 3. Gram's Criterion for Linear Dependence of Vectors

§ 4. Orthogonal Projection

§ 5. The Geometrical Meaning of the Gramian and Some Inequalities

§ 6. Orthogonalization of a Sequence of Vectors

§ 7. Orthonormal Bases

§ 8. The Adjoint Operator

§ 9. Normal Operators in a Unitary Space

§ 10. The Spectra of Normal, Hermitian, and Unitary Operators

§ 11. PositiveSemidefinite and PositiveDefinite Hermitian Operators

§ 12. Polar Decomposition of a Linear Operator in a Unitary Space. Cayley's Formulas

§ 13. Linear Operators in a Euclidean Space

§ 14. Polar Decomposition of an Operator and the Cayley Formulas in a Euclidean Space

§ 15. Commuting Normal Operators

CHAPTER X: QUADRATIC AND HERMITIAN FORMS

§ I. Transformation of the Variables in a Quadratic Form

§ 2. Reduction of a Quadratic Form to a Sum of Squares. The Law of Inertia

§ 3. The Methods of Lagrange and Jacobi of Reducing a Quadratic Form to a Sum of Squares

§ 4. Positive Quadratic Forms

§ 5. Reduction of a Quadratic Form to Principal Axes

§ 6. Pencils of Quadratic Forms

§ 7. Extremal Properties of the Characteristic Values of a Regular Pencil of Forms

§ 8. Small Oscillations of a System with n Degrees of Freedom

§ 9. Hermitian Forms

§ 10. Hankel Forms

BIBLIOGRAPHY

INDEX

Back Cover


Additional Material

Reviews

This is an excellent textbook.
Zentralblatt MATH 
From a review of the original Russian edition ...
The first part (10 chapters; “General theory”) gives in satisfactory detail, and with more than customary completeness, the topics which belong to the main body of the ... subjects ... The point of view is broad and includes much abstract treatment ...
The number of subjects which the book treats well is great ... would appeal to a wide audience.
Mathematical Reviews 
From a review of the English translation ...
The work is an outstanding contribution to matrix theory and contains much material not to be found in any other text.
Mathematical Reviews


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
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This treatise, by one of Russia's leading mathematicians, gives in easily accessible form a coherent account of matrix theory with a view to applications in mathematics, theoretical physics, statistics, electrical engineering, etc. The individual chapters have been kept as far as possible independent of each other, so that the reader acquainted with the contents of Chapter 1 can proceed immediately to the chapters of special interest. Much of the material has been available until now only in the periodical literature.

Front Cover

PREFACE

PUBLISHERS' PREFACE

CONTENTS

CHAPTER I: MATRICES AND OPERATIONS ON MATRICES

§ I. Matrices. Basic Notation

§ 2. Addition and Multiplication of Rectangular Matrices

§ 3. Square Matrices

§ 4. Compound Matrices. Minors of the Inverse Matrix

CHAPTER II: THE ALGORITHM OF GAUSS AND SOME OF ITS APPLICATIONS

§ 1. Gauss's Elimination Method

§ 2. Mechanical Interpretation of Gauss's Algorithm

§ 3. Sylvester's Determinant Identity

§ 4. The Decomposition of a Square Matrix into Triangular Factors

§ 5. The Partition of a Matrix into Blocks. The Technique of Operating with Partitioned Matrices. The Generalized Algorithm of Gauss

CHAPTER III: LINEAR OPERATORS IN AN nDIMENSIONAL VECTOR SPACE

§ I. Vector Spaces

§ 2. A Linear Operator Mapping an nDimensional Space into an mDimensional Space

§ 3. Addition and Multiplication of Linear Operators

§ 4. Transformation of Coordinates

§ 5. Equivalent Matrices. The Rank of an Operator. Sylvester's Inequality

§ 6. Linear Operators Mapping an nDimensional Space into Itself

§ 7. Characteristic Values and Characteristic Vectors of a Linear Operator

§ 8. Linear Operators of Simple Structure

CHAPTER IV: THE CHARACTERISTIC POLYNOMIAL AND THE MINIMAL POLYNOMIAL OF A MATRIX

§ I. Addition and Multiplication of Matrix Polynomials

§ 2. Right and Left Division of Matrix Polynomials

§ 3. The Generalized Bezout Theorem

§ 4. The Characteristic Polynomial of a Matrix. The Adjoint Matrix

§ 5. The Method of Faddeev for the Simultaneous Computation of the Coefficients of the Characteristic Polynomial and of the Adjoint Matrix

§ 6. The Minimal Polynomial of a Matrix

CHAPTER V: FUNCTIONS OF MATRICES

§ I. Definition of a Function of a Matrix

§ 2. The LagrangeSylvester Interpolation Polynomial

§ 3. Other Forms of the Definition of f(A). The Components of the Matrix A

§ 4. Representation of Functions of Matrices by means of Series

§ 5. Application of a Function of a Matrix to the Integration of a System of Linear Differential Equations with Constant Coefficients

§ 6. Stability of Motion in the Case of a Linear System

CHAPTER VI: EQUIVALENT TRANSFORMATIONS OF POLYNOMIAL MATRICES. ANALYTIC THEORY OF ELEMENTARY DIVISORS

§ 1. Elementary Transformations of a Polynomial Matrix

§ 2. Canonical Form of a λMatrix

§ 3. Invariant Polynomials and Elementary Divisors of a Polynomial Matrix

§ 4. Equivalence of Linear Binomials

§ 5. A Criterion for Similarity of Matrices

§ 6. The Normal Forms of a Matrix

§ 7. The Elementary Divisors of the Matrix f(A.)

§ 8. A General Method of Constructing the Transforming Matrix

§ 9. Another Method of Constructing a Transforming Matrix

CHAPTER VII: THE STRUCTURE OF A LINEAR OPERATOR IN AN nDIMENSIONAL SPACE (Geometrical Theory of Elementary Divisors)

§ 1. The Minimal Polynomial of a Vector and a Space (with Respect to a Given Linear Operator)

§ 2. Decomposition into Invariant Subspaces with CoPrime Minimal Polynomials

§ 3. Congruence. Factor Space

§ 4. Decomposition of a Space into Cyclic Invariant Subspaces

§ 5. The Normal Form of a Matrix

§ 6. Invariant Polynomials, Elementary Divisors

§ 7. The Jordan Normal Form of a Matrix

§ 8. K.rylov's Method of Transforming the Secular Equation

CHAPTER VIII: MATRIX EQUATIONS

§ 1. The Equation AX= XB

§ 2. The Special Case .4 = B. Commuting Matrices

§ 3. The Equation AX  XB = C

§ 4. The Scalar Equation f(X) = 0

§ 5. Matrix Polynomial Equations

§ 6. The Extraction of mth Roots of a NonSingular Matrix

§ 7. The Extraction of mth Roots of a Singular Matrix

§ 8. The Logarithm of a Matrix

CHAPTER IX: LINEAR OPERATORS IN A UNITARY SPACE

§ 1. General Considerations

§ 2. Metrization of a Space

§ 3. Gram's Criterion for Linear Dependence of Vectors

§ 4. Orthogonal Projection

§ 5. The Geometrical Meaning of the Gramian and Some Inequalities

§ 6. Orthogonalization of a Sequence of Vectors

§ 7. Orthonormal Bases

§ 8. The Adjoint Operator

§ 9. Normal Operators in a Unitary Space

§ 10. The Spectra of Normal, Hermitian, and Unitary Operators

§ 11. PositiveSemidefinite and PositiveDefinite Hermitian Operators

§ 12. Polar Decomposition of a Linear Operator in a Unitary Space. Cayley's Formulas

§ 13. Linear Operators in a Euclidean Space

§ 14. Polar Decomposition of an Operator and the Cayley Formulas in a Euclidean Space

§ 15. Commuting Normal Operators

CHAPTER X: QUADRATIC AND HERMITIAN FORMS

§ I. Transformation of the Variables in a Quadratic Form

§ 2. Reduction of a Quadratic Form to a Sum of Squares. The Law of Inertia

§ 3. The Methods of Lagrange and Jacobi of Reducing a Quadratic Form to a Sum of Squares

§ 4. Positive Quadratic Forms

§ 5. Reduction of a Quadratic Form to Principal Axes

§ 6. Pencils of Quadratic Forms

§ 7. Extremal Properties of the Characteristic Values of a Regular Pencil of Forms

§ 8. Small Oscillations of a System with n Degrees of Freedom

§ 9. Hermitian Forms

§ 10. Hankel Forms

BIBLIOGRAPHY

INDEX

Back Cover

This is an excellent textbook.
Zentralblatt MATH 
From a review of the original Russian edition ...
The first part (10 chapters; “General theory”) gives in satisfactory detail, and with more than customary completeness, the topics which belong to the main body of the ... subjects ... The point of view is broad and includes much abstract treatment ...
The number of subjects which the book treats well is great ... would appeal to a wide audience.
Mathematical Reviews 
From a review of the English translation ...
The work is an outstanding contribution to matrix theory and contains much material not to be found in any other text.
Mathematical Reviews