The theory of matrices: 1
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2000
|
| Ausgabe: | Reprint. |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | X, 374 S. |
| ISBN: | 0821813765 |
Internformat
MARC
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| 245 | 1 | 0 | |a The theory of matrices |n 1 |c F. R. Gantmacher |
| 250 | |a Reprint. | ||
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2000 | |
| 300 | |a X, 374 S. | ||
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Datensatz im Suchindex
| _version_ | 1804141531968307200 |
|---|---|
| adam_text | CONTENTS
Preface-------------------------------------------------------iii
Publishers’ Preface-------------------------------------------vi
I. Matrices and Operations on Matrices__________________________1
§ 1. Matrices. Basic notation_______________________________ 1
§ 2. Addition and multiplication of rectangular matrices___3
§ 3. Square matrices--------------------------------------12
§ 4. Compound matrices. Minors of the inverse matrix________— 19
II. The Algorithm of Gauss and Some of its Applications 23
§ 1. Gauss’s elimination method-----------------«------------ 23
§ 2. Mechanical interpretation of Gauss’s algorithm-------28
§ 3. Sylvester’s determinant identity_____________________31
§ 4. The decomposition of a square matrix into triangular fac-
tors --------------------..----------------------------------33
§ 5. The partition of a matrix into blocks. The technique of
operating with partitioned matrices. The generalized algo-
rithm of Gauss_________________________________________ 41
III. Linear Operators in an ^-Dimensional Vector Space 50
§ 1. Vector spaces------—___________________________________ 50
§ 2. A linear operator mapping an n-dimensional space into an
m-dimensional space —...w»....»....................... 55
§ 3. Addition and multiplication of linear operators----------57
§ 4. Transformation of coordinates____________________________59
§ 5. Equivalent matrices. The rank of an operator. Sylvester’s
inequality________________________________________ 61
§ 6. Linear operators mapping an n-dimensional space into
itself
66
Contents
Vlll
§ 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 Poly-
nomial of a Matrix__________.____________________________
§ 1. Addition and multiplication of matrix polynomials---
§ 2. Right and left division of matrix polynomials—
§ 3. The generalized Bezout theorem________—
76
76
77
80
§ 4. The characteristic polynomial of a matrix. The adjoint
§ 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___
•« UMM M
95
§
§
1.
2.
3.
4.
5.
§ 6.
Definition of a function of a matrix..
95
101
The Lagrange-Sylvester interpolation polynomial-----
Other forms of the definition of /(A). The components
of the matrix A-----------------------------------------104
Representation of functions of matrices by means of series 110
Application of a function of a matrix to the integration of
a system of linear differential equations with constant
coefficients 116
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 ¿-matrix------------------- 134
§ 3. Invariant polynomials and elementary divisors of a poly-
nomial 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
ix
§ 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 h-Dimen-
sional 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 JSquations™—.—....—™...™..™ 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 /(X) = O______________:----------- 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 in-
equalities 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— •n*m »mm ——mum———
§ 10. The spectra of normal, hermitian, and unitary operators—
§ 11. Positive-semidefinite and positive-definite hermitian op-
erators —
§ 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 for-
mulas in a euclidean space--------------------------------------
§ 15. Commuting normal operators--------------------------------
X. Quadratic and Hermitian Forms______________________________
Transformation of the variables in a quadratic form--
Reduction of a quadratic form to a sum of squares. The
law of inertia
The methods of Lagrange and Jacobi of reducing a quad-
ratic form to a sum of squares-----------------------
Positive quadratic forms_____________________________
Reduction of a quadratic form to principal axes------
Pencils of quadratic forms___________________________
Extremal properties of the characteristic values of a regu-
lar pencil of forms__________________________________
Small oscillations of a system with n degrees of freedom—
Hermitian forms--------------------------------------
Hankel forms_________________________________________
§ l.
§ 2.
§ CO
§ 4.
§ 5.
§ 6.
§ 7.
§ 00
•
§ 9.
§: 10.
Bibliography
Index
268
270
274
276
280
286
290
294
294
296
299
304
308
310
317
326
331
338
351
369
|
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| id | DE-604.BV025690913 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:17:22Z |
| institution | BVB |
| isbn | 0821813765 |
| language | German |
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| physical | X, 374 S. |
| publishDate | 2000 |
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| publisher | American Mathematical Society |
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| spelling | Gantmacher, Feliks R. 1908-1964 Verfasser (DE-588)129584703 aut Teorija matric The theory of matrices 1 F. R. Gantmacher Reprint. Providence, RI American Mathematical Society 2000 X, 374 S. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke (DE-604)BV025384231 1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019295245&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gantmacher, Feliks R. 1908-1964 The theory of matrices |
| title | The theory of matrices |
| title_alt | Teorija matric |
| title_auth | The theory of matrices |
| title_exact_search | The theory of matrices |
| title_full | The theory of matrices 1 F. R. Gantmacher |
| title_fullStr | The theory of matrices 1 F. R. Gantmacher |
| title_full_unstemmed | The theory of matrices 1 F. R. Gantmacher |
| title_short | The theory of matrices |
| title_sort | the theory of matrices |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019295245&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV025384231 |
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