Isotropic Basis in a Real Symmetric Bilinear Space: Existence and Construction
DOI:
https://doi.org/10.52737/18291163-2025.17.14-1-19Keywords:
Symmetric Bilinear Form, Symmetric Bilinear Space, Isotropic Basis, Preimage DimensionAbstract
A basis of a real symmetric bilinear space is called an isotropic basis if all its elements are isotropic. In this paper, we provide both necessary and sufficient conditions for the existence of such an isotropic basis. We present one geometric method and two linear algebraic methods for constructing isotropic bases. Additionally, we address a question arising from the properties of symmetric bilinear forms. As a consequence, we explore various properties of the vector space spanned by the preimage set of a point under a real-valued continuous function. We also demonstrate some applications of these properties within the context of real symmetric bilinear spaces.
References
Z.Z. Bai and J.Y. Pan, Matrix Analysis and Computations, SIAM, 2021. DOI: https://doi.org/10.1137/1.9781611976632
M. Berger, Geometry I,II, Translated from the French by M. Cole and S. Levy, Corrected 4th printing, Springer, Berlin/Heidelberg/New York, 2009.
D. Biswas and S. Dutta, Möbius action of SL(2; R) on different homogeneous spaces. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 92 (2022), no. 1, pp. 23-29. DOI: https://doi.org/10.1007/s40010-020-00673-1
D. Biswas and S. Dutta, Invariant projective properties under the action of the Lie group SL(3;R) on RP2. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 93 (2023), no. 2, pp. 301-314. DOI: https://doi.org/10.1007/s40010-023-00813-3
D. Biswas and I. Rajwar, Projective disk and the action of SL(3, R): Exploring orthogonality. Lobachevskii J. Math., 44 (2023), no. 12, pp. 5193-5203. DOI: https://doi.org/10.1134/S1995080223120090
D. Biswas and I. Rajwar, Projcetive disk and Poincaré disk: Exploring SL(3,R) action from an Erlangen perspective. Results Math., 80 (2025), article number 109. DOI: https://doi.org/10.1007/s00025-025-02417-2
N. Bourbaki, General Topology: Chapter 5-10, 1st ed., Springer-Verlag, Berlin/Heidelberg, 1995. DOI: https://doi.org/10.1007/978-3-642-61701-0
T.E. Cecil, Lie Sphere Geometry, 2nd ed., Universitext, Springer, New York, 2008.
P.B. Gilkey, Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor, World Scientific, Singapore, 2020.
W. Greub, Linear Algebra, Springer-Verlag, New York, 1975. DOI: https://doi.org/10.1007/978-1-4684-9446-4
A.A. Kirillov, A Tale of Two Fractals, 1st ed., Birkhäuser New York, Springer, 2013. DOI: https://doi.org/10.1007/978-0-8176-8382-5
V.V. Kisil, Geometry of Möbius Transformations, Imperial College Press, London, 2012. DOI: https://doi.org/10.1142/p835
V.V. Kisil, Cycles cross ratio: An invitation. Elem. Math., 78 (2023), no. 2, pp. 49-71. DOI: https://doi.org/10.4171/em/471
T.Y. Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society, 2005. DOI: https://doi.org/10.1090/gsm/067
J.R. Munkres, Topology, 2nd ed., Upper Saddle River, NJ: Prentice Hall, 2000.
G.L. Naber, The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity, 2nd ed., Springer, New York, 2012. DOI: https://doi.org/10.1007/978-1-4419-7838-7
A.R. Rao and P. Bhimasankaram, Linear Algebra, 2nd ed., Hindustan Book Agency, New Delhi, 2000. DOI: https://doi.org/10.1007/978-93-86279-01-9
W. Scharlau, Quadratic and Hermitian Forms, 1st ed., Springer Berlin, Heidelberg, 1985. DOI: https://doi.org/10.1007/978-3-642-69971-9_1
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Armenian Journal of Mathematics
This work is licensed under a Creative Commons Attribution 4.0 International License.
