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Matrizentheorie

Matrizentheorie


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About the Book

12.1. 1. In diesem Kapitel wird folgende Frage behandelt: Gegeben seien vier Matnzen A, B, A1, B1 gleichen Typs (m, n) mit Elementen aus e nem Zahlkorper K. Gesucht s nd die Bedingungen, unter denen zwei regulare quadra- t 8che Matrizen P und Q der Ordnung m bzw. n existieren derart, dafJ gleichzeitig (1) giU. 1) Fuhrt man die Matrizenbuschel A + J..B und A1 + J..B ein, so k6nnen die beiden 1 Matrizengleichungen (1) durch die einzige Gleichung (2) P(A + J..B) Q = A1 + J..B1 ersetzt werden. Definition 1. Wir nennen zwei Buschel A + J..B und A1 + J..B rechteckiger Ma- 1 trizen gleichen Typs (m, n) streng aquivalent, wenn fUr sie die Gleichung (2) gilt und dabei P und Q konstante (d. h. von J.. unabhiingige) regulare quadratische Matrizen 2 (m-ter bzw. n-ter Ordnung) sind. ) Nach der allgemeinen Definition, der Aquivalenz von Polynommatrizen (vgl.

Table of Contents:
Erster Teil: Allgemeine Theorie.- 1. Matrizen und Matrizenoperationen.- 2. Der Gaußsche Algorithmus.- 3. Lineare Operatoren im n-dimensionalen Vektorraum.- 4. Charakteristisches Polynom und Minimalpolynom einer Matrix.- 5. Matrizenfunktionen.- 6. Äquivalente Transformationen von Polynommatrizen. Analytische Elementarteilertheorie.- 7. Die Struktur linearer Operatoren im n-dimensionalen Vektorraum. Geometrische Elementarteilertheorie.- 8. Matrizengleichungen.- 9. Lineare Operatoren im unitären Raum.- 10. Quadratische und hermitesche Formen.- Zweiter Teil: Spezielle Fragen und Anwendungen.- 11. Komplexe symmetrische, schief symmetrische und orthogonale Matrizen.- 12. Singuläre Matrizenbüschel.- 13. Matrizen mit nichtnegativen Elementen.- 14. Verschiedene Regularitätskriterien und die Lokalisierung der charakteristischen Wurzeln.- 15. Anwendungen der Matrizenrechnung zur Untersuchung linearer Differentialgleichungssysteme.- 16. Das Routh-Hurwitzsehe Problem und verwandte Fragen.- Anhang von V. B. Lidskij.- Ungleichungen für charakteristische und singuläre Wurzeln.- 1. Majorantenfolgen.- 2. Die Horn-Neumannschen Ungleichungen.- 3. Die Weylschen Ungleichungen.- 4. Maximal-Minimaleigenschaften von Summen und Produkten der charakteristischen Wurzeln hermitescher Operatoren.- 5. Ungleichungen für charakteristische und singuläre Wurzeln von Operatorsumnien und -produkten.- 6. Eine andere Aufgabenstellung bezüglich des Spektrums von Summen und Produkten hermitescher Operatoren.- Literatur.- Namen- und Sachverzeichnis.


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Product Details
  • ISBN-13: 9783642712449
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
  • Publisher Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Height: 244 mm
  • No of Pages: 654
  • Returnable: N
  • ISBN-10: 3642712444
  • Publisher Date: 17 Nov 2011
  • Binding: Paperback
  • Language: German
  • Returnable: N
  • Width: 170 mm


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