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2 × 2 Matrices That Are Roots of Unity

  • Alexander A. Roytvarf
Chapter

Abstract

As readers know, a polynomial equation of degree n has at most n roots (considering multiplicity) in a number field containing its coefficients. How many roots does a polynomial equation have in a matrix ring? First, how many roots of degree n of 1=E, that is, matrices X, \(X^n = {X \circ \ldots \circ X}_n = E\) are there? And how would one enumerate them? These and related questions that can be answered given a relatively modest level of knowledge on the reader’s part are discussed in this chapter. That is, we will characterize the roots of E in a ring of 2×2 matrices with real entries and show some applications to other mathematical topics: matrix norm estimation and spectral analysis of 3-diagonal Jacobi matrices.

Keywords

Linear Operator Problem Group Normed Vector Space Jordan Canonical Form Trigonometric Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Arnol’d, V.I.: Обыкновенные дифференциальные уравнения. “Наука” Press, Moscow (1975). [English transl. Ordinary Differential Equations. MIT Press, Cambridge (1978)]Google Scholar
  2. Arnol’d, V.I.: Дополнительные главы теории обыкновенных дифференциальных уравнений. “Наука” Press, Moscow (1978). Геометрические методы в теории обыкновенных дифференциальных уравнений. Regular & Chaotic Dynamics, 2nd edn. MCNMO/VMK NMU (1999). [English transl. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, Berlin/Heidelberg (2004)]Google Scholar
  3. Beckenbach, E.F., Bellman, R.: Inequalities. Springer, Berlin/Göttingen/Heidelberg (1961)CrossRefGoogle Scholar
  4. Belitskii, G.R., Lubich, Yu.I.: Нормы матриц и их приложения. “Наукова Думка” Press, Kiev (1984). [English transl. Matrix Norms and their Applications (Operator Theory Advance and Applications, V. 36). Springer (1988)]Google Scholar
  5. Bellman, R.: Introduction to Matrix Analysis. McGraw-Hill, New York/Toronto/London (1960)MATHGoogle Scholar
  6. Gantmacher, F.R.: Теория матриц, 3rd edn. “Наука” Press, Moscow (1967). [English transl. Theory of Matrices. V.’s 1–2. Chelsea (2000)]Google Scholar
  7. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin/Heidelberg/New York/Tokyo (1983)MATHCrossRefGoogle Scholar
  8. Robinson, A., Roytvarf, A.: Computerized tomography for non-destructive testing. EU 99971130, 2–2218 (2001)Google Scholar
  9. Schmidt, W.M.: In: Dold, A., Eckmann, B. (eds.) Diophantine Approximations. Lecture Notes in Mathematics, vol. 785. Springer, Berlin/Heidelberg/New York (1980)Google Scholar
  10. Trigg, C.: Mathematical Quickies. McGraw-Hill, New York/London (1967)Google Scholar
  11. Venkataraman, C.S.: In: Problems and solutions. Math. Magazine 44(1), 55 (1971)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Alexander A. Roytvarf
    • 1
  1. 1.Rishon LeZionIsrael

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