Certain nonlinear functions acting on the vector space \(\mathbb{H}^{n}\) over the Quaternions \(\mathbb{H}\)
Authors
Ilwoo Cho
- Dept. of Math. and Stat., 421 Ambrose Hall, St. Ambrose Univ., 518 W. Locust St., Davenport, Iowa, 52803, USA.
Abstract
In this paper, we consider a certain type of nonlinear functions acting
on a finite-dimensional vector space \(\mathbb{H}^{n}\) over the ring
\(\mathbb{H}\) of all quaternions, for \(n\) \(\in\) \(\mathbb{N}.\) Our
main results show that: (i) every quaternion \( {q\in\mathbb{H}}\)
is classified by its spectrum of the realization under a canonical
representation on \(\mathbb{C}^{2}\); (ii) each vector of \(\mathbb{H}^{n}\)
is classified by \(\mathbb{C}^{n}\) in an extended set-up of (i); and
(iii) the (usual linear) spectral analysis on the matricial ring \( {M_{n}\left(\mathbb{C}\right)}\)
of all \(\left(n\times n\right)\)-matrices (over \(\mathbb{C}\)) affects
some fixed point theorems for our nonlinear functions on \(\mathbb{H}^{n}\).
In conclusion, we study the connections between the ``linear'' spectral
theory over the complex numbers \(\mathbb{C}\), and fixed point theorems
for ``nonlinear'' functions over \(\mathbb{H}\).
Share and Cite
ISRP Style
Ilwoo Cho, Certain nonlinear functions acting on the vector space \(\mathbb{H}^{n}\) over the Quaternions \(\mathbb{H}\), Journal of Nonlinear Sciences and Applications, 15 (2022), no. 1, 14--40
AMA Style
Cho Ilwoo, Certain nonlinear functions acting on the vector space \(\mathbb{H}^{n}\) over the Quaternions \(\mathbb{H}\). J. Nonlinear Sci. Appl. (2022); 15(1):14--40
Chicago/Turabian Style
Cho, Ilwoo. "Certain nonlinear functions acting on the vector space \(\mathbb{H}^{n}\) over the Quaternions \(\mathbb{H}\)." Journal of Nonlinear Sciences and Applications, 15, no. 1 (2022): 14--40
Keywords
- The quaternions
- q-spectral forms
- q-spectralizations
- vector spaces over the quaternions
MSC
References
-
[1]
I. Cho, A Spectral Representation of the Quaternions, Submitted to Comput. Methods & Funct. Theo., 2020 (2020), Preprint
-
[2]
I. Cho, P. E. T. Jorgensen, Spectral Analysis of Equations over Quaternions, Submitted to Internat. Conf. Stochastic Processes Alge. Structures: From Theory towards Appl. (SPAS 2019) Vestera, Sweden, Published by Springer, 2020 (2020), Preprint
-
[3]
C. Doran, A. Lasenby, Geometric Algebra for Physicists, Cambridge University Press, Cambridge (2003)
-
[4]
F. O. Farid, Q.-W. Wang, F. Zhang, On the Eigenvalues of Quaternion Matrices, Linear Multilinear Algebra, 59 (2011), 451--473
-
[5]
C. Flaut, Eigenvalues and eigenvectors for the quaternion matrices of degree two, An. S¸ tiint¸. Univ. Ovidius Constant¸a Ser. Mat., 10 (2002), 39--44
-
[6]
P. R. Girard, Einstein’s equations and Clifford algebra, Adv. Appl. Clifford Algebras, 9 (1999), 225--230
-
[7]
P. R. Halmos, A Hilbert space problem book, Springer-Verlag, New York (1982)
-
[8]
P. R. Halmos, Linear Algebra Problem Book, Mathematical Association of America, Washington, DC (1995)
-
[9]
W. R. Hamilton, Lectures on Quaternions, Hodges & Smith, Dublin (1853)
-
[10]
I. L. Kantor, A. S. Solodnikov, Hypercomplex Numbers: an Elementary Introuction to Algebras, Springer-Verlag, New York (1989)
-
[11]
V. V. Kravchenko, Applied Quaternionic Analysis, Heldermann Verlag, Lemgo (2003)
-
[12]
S. D. Leo, G. Scolarici, L. Solombrino, Quaternionic Eigenvalue Problem, J. Math. Phys., 43 (2002), 5815--5829
-
[13]
T. Li, Eigenvalues and eigenvectors of quaternion matrices, J. Central China Normal Univ. Natur. Sci., 29 (1995), 407--411
-
[14]
N. Mackey, Hamilton and Jacobi meet again: quaternions and the eigenvalue problem, SIAM J. Matrix Anal. Appl.,, 16 (1995), 421--435
-
[15]
S. Qaisar, L. Zou, Distribution for the Standard Eigenvalues of Quaternion Matrices, Int. Math. Forum, 7 (2012), 831--838
-
[16]
L. Rodman, Topics in Quaternion Linear Algebra, Princeton University Press, Princeton (2014)
-
[17]
B. A. Rosenfeld, A history of non-Euclidean geometry: evolution of the concept of a geometric space, Springer-Verlag, New York (1988)
-
[18]
A. Sudbery, Quaternionic Analysis, Math. Proc. Cambridge Philos. Soc., 85 (1979), 199--225
-
[19]
J. Vince, Geometric algebra for computer graphics, Springer-Verlag, London (2008)
-
[20]
J. Voight, Quaternion Algebras, online version, (2019)
