INSEAD Library

Includes bibliographical references and index

Vol. 1, 374 p. ; Vol. 2, 276 p.

Digitized

The Theory of Matrices Volume One Contents PREFACE. PUBLISHER'S PREFACE iii vi I. MATRICES AND OPERATIONS ON MATRICES § 1 Matrices. Basic notation § 2 Addition and multiplication of rectangular matrices § 3 Square matrices § 4 Compound matrices. Minors of the inverse 1 1 3 19 II. THE ALGORITHM OF GAUSS AND SOME OF ITS APPLICATIONS 23 § 1. Gauss's elimination § 2. Mechanical interpretation of Gauss's § 3. Sylvester's determinant identity § 4. The decomposition of a square matrix into triangular factors 23 28 31 33 § 5. The partition of a matrix into blocks. The technique of operating with partitioned matrices. The generalized algorithm of Gauss 41 III. LINEAR OPERATORS IN AN n-DIMENSIONAL VECTOR SPACE § 1. Vector spaces 50 50 55 57 59 61 66 § 2. A linear operator mapping an n-dimensional space into an m-dimensional space § 3. Addition and multiplication of linear § 4. Transformation of coordinates § 5. Equivalent matrices. The rank of an operator. Sylvester's inequality § 6. Linear operators mapping an n-dimensional space into itself viii CONTENTS § 7. Characteristic values and characteristic vectors of a linear operator ............................................................................... 69 § 8. Linear operators of simple structure..................................... 72 IV. THE CHARACTERISTIC POLYNOMIAL AND THE MINIMAL POLYNOMIAL OF A MATRIX................................................................................ 76 § 1. Addition and multiplication of matrix polynomials................ 76 § 2. Right and left division of matrix polynomials......................... 77 § 3. The generalized Bezout theorem.............................................. 80 § 4. The characteristic polynomial of a matrix. The adjoint matrix .................................................................................. 82 § 5. The method of Faddeev for the simultaneous computation of the coefficients of the characteristic polynomial and of the adjoint matrix ............................................................... 87 § 6. The minimal polynomial of a matrix....................................... 89 V. FUNCTIONS OF MATRICES........................................................................ 95 § 1. Definition of a function of a matrix........................................ 95 § 2. The Lagrange-Sylvester interpolation polynomial................ 101 § 3. Other forms of the definition of f (A). The components of the matrix A................................................................................. 104 § 4. Representation of functions of matrices by means of series 110 § 5. Application of a function of a matrix to the integration of a system of linear differential equations with constant coefficients ............................................................................ 116 § 6. Stability of motion in the case of a linear system................ 125 VI. EQUIVALENT TRANSFORMATIONS OF POLYNOMIAL MATRICES ANALYTIC THEORY OF ELEMENTARY DIVISORS.......................... 130 § 1. Elementary transformations of a polynomial matrix........... 130 § 2. Canonical form of a 2-matrix................................................ 134 § 3. Invariant polynomials and elementary divisors of a polynomial matrix .................................................................... 139 § 4. Equivalence of linear binomials............................................ 145 § 5. A criterion for similarity of matrices..................................... 147 § 6. The normal forms of a matrix............................................... 149 § 7. The elementary divisors of the matrix f (A)................................. 153 CONTENTS 1X § 8. A general method of constructing the transforming matrix 159 § 9. Another method of constructing a transforming matrix........ 164 VII. THE STRUCTURE OF A LINEAR OPERATOR IN AN n-DIMENSIONAL SPACE............................................................................................................. 175 § 1. The minimal polynomial of a vector and a space (with respect to a given linear operator)....................................................... 175 § 2. Decomposition into invariant subspaces with co-prime minimal polynomials .......................................................... 177 § 3. Congruence. Factor space...................................................... 181 § 4. Decomposition of a space into cyclic invariant subspaces.... 184 § 5. The normal form of a matrix................................................... 190 § 6. Invariant polynomials. Elementary divisors........................... 193 § 7. The Jordan normal form of a matrix...................................... 200 § 8. Krylov's method of transforming the secular equation.......... 202 VIII. MATRIX EQUATIONS. ............................................................................... 215 § 1. The equation AX=XB.............................................................................. 215 § 2. The special case A=B. Commuting matrices.......................... 220 § 3. The equation AX -- XB = C............................................................. 225 § 4. The scalar equation f(X) = 0............................................................ 225 § 5. Matrix polynomial equations .................................................. 227 § 6. The extraction of m-th roots of a non-singular matrix........... 231 § 7. The extraction of m-th roots of a singular matrix.................. 234 § 8. The logarithm of a matrix........................................................ 239 IX. LINEAR OPERATORS IN A UNITARY SPACE........................................ 242 § 1. General considerations........................................................... 242 § 2. Metrization of a space............................................................. 243 § 3. Gram's criterion for linear dependence of vectors................. 246 § 4. Orthogonal projection ............................................................. 248 § 5. The geometrical meaning of the Gramian and some inequalities ........................................................................... 250 § 6. Orthogonalization of a sequence of vectors............................ 256 § 7. Orthonormal bases ............................................................... 262 § 8. The adjoint operator ............................................................... 265 X CONTENTS § 9. Normal operators in a unitary space... 268 § 10. The spectra of normal, hermitian, and unitary operators-- 270 § 11. Positive-semidefinite and positive-definite hermitian operator 271 § 12. Polar decomposition of a linear operator in a unitary space Cayley's formulas § 13. Linear operators in a euclidean 276 280 § 14. Polar decomposition of an operator and the Cayley for mulas in a euclidean space ...................................... ...... 286 § 15. Commuting normal operators ................................. .. ..... 290 X. QUADRATIC AND HERMITIAN § 1. Transformation of the variables in a quadratic form........ 294 294 § 2. Reduction of a quadratic form to a sum of squares. The law of inertia 296 § 3. The methods of Lagrange and Jacobi of reducing a quad ratic form to a sum of squares 299 § 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 regu la pencil of forms 317 304 308 304 § 8. Small oscillations of a system with n degrees of freedom...... 326 § 9. Hermitian forms 331 BIBLIOGRAPHY INDEX 351 369 The Theory of Matrices Volume Two Contents PREFACE PUBLISHERS' PREFACE iii vi XI. COMPLEX SYMMETRIC, SKEW-SYMMETRIC, AND ORTHOGONAL MATRICES §1 Some formulas for complex orthogonal and unitary matrices §2 Polar decomposition of a complex matrix........... ............. § 3 The normal form of a complex symmetric matrix....................... 9 §. 4. The normal form of a complex skew-symmetric matrix 12 § 5. The normal form of a complex orthogonal matrix................... 18 XII. SINGULAR PENCILS OF § 1 Introduction § 2 Regular pencils of matrices § 3 Singular pencils. The reduction theorem...... § 4 The canonical form of a singular pencil of matrices § 5 The minimal indices of a pencil. Criterion for strong equivalence of pencils § 6 Singular pencils of quadratic forms 37 40 24 24 25 29 35 1 6 § 7 Application to differential equations........................................... 45 XIII. MATRICES WITH NON-NEGATIVE ELEMENTS § 1 General properties § 2 Spectral properties of irreducible non-negative matrices. § 3 Reducible matrices § 4 The normal form of a reducible matrix § 5 Primitive and imprimitive matrices § 6 Stochastic matrices 50 50 53 66 74 80 82 viii CONTENTS § 7. Limiting probabilities for a homogeneous Markov chain with a finite number of......... § 8. Totally non-negative matrices .... ....... ...... § 9. Oscillatory matrices 87 98 103 XIV. APPLICATIONS OF THE THEORY OF MATRICES TO THE INVESTIGA TION OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 113 § 1. Systems of linear differential equations with variable coefficients. General concepts 113 § 2. Lyapunov transformations ..................................................... 116 § 3. Reducible systems § 5. The matricant § 6. The multiplicative integral. The infinitesimal calculus of Volterra ........... 118 125 131 § 4. The canonical form of a reducible system. Erugin's theorem 121 § 7. Differential systems in a complex domain. General properties 135 § 8. The multiplicative integral in a complex domain................... 138 § 9. Isolated singular points . § 10. Regular singularities ................................... § 11. Reducible analytic systems . 142 148 164 § 12. Analytic functions of several matrices and their applica tion to the investigation of differential systems. The papers of Lappo-Danilevskii.................................................................. 168 XV. THE PROBLEM OF ROUTH-HURWITZ AND RELATED QUESTIONS 172 § 1. Introduction .......................................................................... 172 § 2. Cauchy indices § 4. The singular case. Examples ...... § 5. Lyapunov 's theorem 173 181 185 § 3. Routh 's algorithm ................................................................ 177 § 6. The theorem of Routh-Hurwitz................................................ 190 § 7. Orlando's formula ................................................................ 196 § 8. Singular cases in the Routh-Hurwitz theorem...................... 198 § 9. The method of quadratic forms. Determination of the number of distinct real roots of a polynomial..................... 201 CONTENTS ix § 10. Infinite Hankel matrices of finite rank. ............................... 204 § 11. Determination of the index of an arbitrary rational fraction by the coefficients of numerator and denominator..... 208 § 12. Another proof of the Routh-Hurwitz theorem...................... 216 § 13. Some supplements to the Routh-Hurwitz theorem. Stability criterion of Liénard and Chipart.. 220 § 14. Some properties of Hurwitz polynomials. Stieltjes' theorem. Representation of Hurwitz polynomials by continued fractions 225 § 15. Domain of stability. Markov parameters.............................. 232 § 16. Connection with the problem of moments...... ................... 236 § 17. Theorems of Markov and Chebyshev.................................... 240 § 18. The generalized Routh-Hurwitz problem............................... 248 BIBLIOGRAPHY INDEX 251 268