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Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity

Year 2021, , 147 - 153, 30.12.2021
https://doi.org/10.32323/ujma.983509

Abstract

In this paper pairs of matrices under similarity are considered because of their scientific applications, especially pairs of matrices being simultaneously diagonalizable. For example, a problem in quantum mechanics is the position and momentum operators, because they do not have a shared base representing the system's states. They do not commute, and that is why switching operators form a crucial element in quantum physics. A study of the set of linear operators consisting of pairs of simultaneously diagonalizable matrices is done using geometric constructions such as the principal bundles. The main goal of this work is to construct connections that allow us to establish a relationship between the local geometry around a point with the local geometry around another point. The connections give us a way to help distinguish bundle sections along tangent vectors.

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References

  • [1] A.A. Albert, B. Muckenhoupt, On matrices of trace zero, Michigan Math. J. , 4 (1957), 1–3.
  • [2] V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26(2) (1971), 29–43.
  • [3] H.J. Bernstein, A.V. Phillips, Fiber bundles and quantum theory, Scientific American 245(1) (1981), 122–137.
  • [4] M.L.A. Flores, Espacios Fibrados, Clases Carater´ısticas y el Isomorfismo de Thom. Pontificia Universidad Cat´olica del Peru-CENTRUM Catolica (Peru), (2013)
  • [5] Sh. Friedland, Simultaneous Similarity of Matrices, Adv. Math., 50 (1983), 189–265.
  • [6] F. Gaines, A Note on Matrices with Zero Trace Amer. Math. Month., 73(6) (1966), 630–631.
  • [7] M.I Garcia-Planas, On simultaneously and approximately simultaneously diagonalizable pairs of matrices Fundam. J. Math. Appl., 2 (2019), 50–55.
  • [8] M.I. Garcia-Planas, T. Klymchuk, Differentiable families of traceless matrix triples RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat, 114 (2019), 1–8.
  • [9] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge, 2013.
  • [10] D. Husemoller. Fibre bundles (Vol. 5). McGraw-Hill, New York, 1966.
  • [11] E. Lubkin, Geometric definition of gauge invariance, Ann. Physics, 23(2) (1963), 233-283.
  • [12] J.M. Maillard, F.Y. Wu, C.K. Hu, Thermal transmissivity in discrete spin systems: Formulation and applications. J. Physics A: Math. Gen., 25(9) (1992), 2521.
  • [13] S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press. 1995.
  • [14] A. Trautman, Fiber bundles, gauge fields, and gravitation. General relativity and gravitation 1 (1980), 287–308.
Year 2021, , 147 - 153, 30.12.2021
https://doi.org/10.32323/ujma.983509

Abstract

References

  • [1] A.A. Albert, B. Muckenhoupt, On matrices of trace zero, Michigan Math. J. , 4 (1957), 1–3.
  • [2] V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26(2) (1971), 29–43.
  • [3] H.J. Bernstein, A.V. Phillips, Fiber bundles and quantum theory, Scientific American 245(1) (1981), 122–137.
  • [4] M.L.A. Flores, Espacios Fibrados, Clases Carater´ısticas y el Isomorfismo de Thom. Pontificia Universidad Cat´olica del Peru-CENTRUM Catolica (Peru), (2013)
  • [5] Sh. Friedland, Simultaneous Similarity of Matrices, Adv. Math., 50 (1983), 189–265.
  • [6] F. Gaines, A Note on Matrices with Zero Trace Amer. Math. Month., 73(6) (1966), 630–631.
  • [7] M.I Garcia-Planas, On simultaneously and approximately simultaneously diagonalizable pairs of matrices Fundam. J. Math. Appl., 2 (2019), 50–55.
  • [8] M.I. Garcia-Planas, T. Klymchuk, Differentiable families of traceless matrix triples RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat, 114 (2019), 1–8.
  • [9] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge, 2013.
  • [10] D. Husemoller. Fibre bundles (Vol. 5). McGraw-Hill, New York, 1966.
  • [11] E. Lubkin, Geometric definition of gauge invariance, Ann. Physics, 23(2) (1963), 233-283.
  • [12] J.M. Maillard, F.Y. Wu, C.K. Hu, Thermal transmissivity in discrete spin systems: Formulation and applications. J. Physics A: Math. Gen., 25(9) (1992), 2521.
  • [13] S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press. 1995.
  • [14] A. Trautman, Fiber bundles, gauge fields, and gravitation. General relativity and gravitation 1 (1980), 287–308.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Maria İsabel Garcia-planas

Publication Date December 30, 2021
Submission Date August 16, 2021
Acceptance Date November 19, 2021
Published in Issue Year 2021

Cite

APA Garcia-planas, M. İ. (2021). Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity. Universal Journal of Mathematics and Applications, 4(4), 147-153. https://doi.org/10.32323/ujma.983509
AMA Garcia-planas Mİ. Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity. Univ. J. Math. Appl. December 2021;4(4):147-153. doi:10.32323/ujma.983509
Chicago Garcia-planas, Maria İsabel. “Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity”. Universal Journal of Mathematics and Applications 4, no. 4 (December 2021): 147-53. https://doi.org/10.32323/ujma.983509.
EndNote Garcia-planas Mİ (December 1, 2021) Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity. Universal Journal of Mathematics and Applications 4 4 147–153.
IEEE M. İ. Garcia-planas, “Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity”, Univ. J. Math. Appl., vol. 4, no. 4, pp. 147–153, 2021, doi: 10.32323/ujma.983509.
ISNAD Garcia-planas, Maria İsabel. “Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity”. Universal Journal of Mathematics and Applications 4/4 (December 2021), 147-153. https://doi.org/10.32323/ujma.983509.
JAMA Garcia-planas Mİ. Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity. Univ. J. Math. Appl. 2021;4:147–153.
MLA Garcia-planas, Maria İsabel. “Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity”. Universal Journal of Mathematics and Applications, vol. 4, no. 4, 2021, pp. 147-53, doi:10.32323/ujma.983509.
Vancouver Garcia-planas Mİ. Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity. Univ. J. Math. Appl. 2021;4(4):147-53.

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