Theory of Matrices, Volume 1

This is an excellent textbook."" — Zentralblatt MATHFrom 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

• Front CoverPREFACEPUBLISHERS'PREFACECONTENTSCHAPTER I: MATRICES AND OPERATIONS ON MATRICESI. Matrices. Basic Notation2. Addition and Multiplication of Rectangular Matrices3. Square Matrices4. Compound Matrices. Minors of the Inverse MatrixCHAPTER II: THE ALGORITHM OF GAUSS AND SOME OF ITS APPLICATIONS1. Gauss's Elimination Method2. Mechanical Interpretation of Gauss's Algorithm3. Sylvester's Determinant Identity4. The Decomposition of a Square Matrix into Triangular Factors5. The Partition of a Matrix into Blocks. The Technique of Operating with Partitioned Matrices. The Generalized Algorithm of GaussCHAPTER III: LINEAR OPERATORS IN AN n-DIMENSIONAL VECTOR SPACEI. Vector Spaces2. A Linear Operator Mapping an n-Dimensional Space into an m-Dimensional Space3. Addition and Multiplication of Linear Operators4. Transformation of Coordinates5. Equivalent Matrices. The Rank of an Operator. Sylvester's Inequality6. Linear Operators Mapping an n-Dimensional Space into Itself7. Characteristic Values and Characteristic Vectors of a Linear Operator8. Linear Operators of Simple StructureCHAPTER IV: THE CHARACTERISTIC POLYNOMIAL AND THE MINIMAL POLYNOMIAL OF A MATRIXI. Addition and Multiplication of Matrix Polynomials2. Right and Left Division of Matrix Polynomials3. The Generalized Bezout Theorem4. The Characteristic Polynomial of a Matrix. The Adjoint Matrix5. The Method of Faddeev for the Simultaneous Computation of the Coefficients of the Characteristic Polynomial and of the Adjoint Matrix6. The Minimal Polynomial of a MatrixCHAPTER V: FUNCTIONS OF MATRICESI. Definition of a Function of a Matrix2. The Lagrange-Sylvester Interpolation Polynomial3. Other Forms of the Definition of f(A). The Components of the Matrix A4. Representation of Functions of Matrices by means of Series5. Application of a Function of a Matrix to the Integration of a System of Linear Differential Equations with Constant Coefficients6. Stability of Motion in the Case of a Linear SystemCHAPTER VI: EQUIVALENT TRANSFORMATIONS OF POLYNOMIAL MATRICES. ANALYTIC THEORY OF ELEMENTARY DIVISORS1. Elementary Transformations of a Polynomial Matrix2. Canonical Form of a λ-Matrix3. Invariant Polynomials and Elementary Divisors of a Polynomial Matrix4. Equivalence of Linear Binomials5. A Criterion for Similarity of Matrices6. The Normal Forms of a Matrix7. The Elementary Divisors of the Matrix f(A.)8. A General Method of Constructing the Transforming Matrix9. Another Method of Constructing a Transforming MatrixCHAPTER VII: THE STRUCTURE OF A LINEAR OPERATOR IN AN n-DIMENSIONAL 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 Co-Prime Minimal Polynomials3. Congruence. Factor Space4. Decomposition of a Space into Cyclic Invariant Subspaces5. The Normal Form of a Matrix6. Invariant Polynomials, Elementary Divisors7. The Jordan Normal Form of a Matrix8. K.rylov's Method of Transforming the Secular EquationCHAPTER VIII: MATRIX EQUATIONS1. The Equation AX= XB2. The Special Case .4 = B. Commuting Matrices3. The Equation AX - XB = C4. The Scalar Equation f(X) = 05. Matrix Polynomial Equations6. The Extraction of m-th Roots of a Non-Singular Matrix7. The Extraction of m-th Roots of a Singular Matrix8. The Logarithm of a MatrixCHAPTER IX: LINEAR OPERATORS IN A UNITARY SPACE1. General Considerations2. Metrization of a Space3. Gram's Criterion for Linear Dependence of Vectors4. Orthogonal Projection5. The Geometrical Meaning of the Gramian and Some Inequalities6. Orthogonalization of a Sequence of Vectors7. Orthonormal Bases8. The Adjoint Operator9. Normal Operators in a Unitary Space10. The Spectra of Normal, Hermitian, and Unitary Operators11. Positive-Semidefinite and Positive-Definite Hermitian Operators12. Polar Decomposition of a Linear Operator in a Unitary Space. Cayley's Formulas13. Linear Operators in a Euclidean Space14. Polar Decomposition of an Operator and the Cayley Formulas in a Euclidean Space15. Commuting Normal OperatorsCHAPTER X: QUADRATIC AND HERMITIAN FORMSI. Transformation of the Variables in a Quadratic Form2. Reduction of a Quadratic Form to a Sum of Squares. The Law of Inertia3. The Methods of Lagrange and Jacobi of Reducing a Quadratic Form to a Sum of Squares4. Positive Quadratic Forms5. Reduction of a Quadratic Form to Principal Axes6. Pencils of Quadratic Forms7. Extremal Properties of the Characteristic Values of a Regular Pencil of Forms8. Small Oscillations of a System with n Degrees of Freedom9. Hermitian Forms10. Hankel FormsBIBLIOGRAPHYINDEXBack Cover