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The Theory of matrices

Author: Gantmacher, F. R. ; Hirsch, K. A., translation Series: AMS-Chelsea book series ; 131 ; 133 Publisher: AMS Chelsea Publishing, 1977.Language: EnglishDescription: 374 p. ; 24 cm.ISBN: 0821813765Type of document: BookBibliography/Index: Includes bibliographical references and indexContents Note: Vol. 1, 374 p. ; Vol. 2, 276 p.
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Europe Campus
Main Collection
Print QA155 .G36 1977 Vol.1
(Browse shelf)
001230717
Available 001230717
Book Europe Campus
Main Collection
Print QA155 .G36 1988 Vol.2
(Browse shelf)
001230725
Available 001230725
Total holds: 0

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

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