Introducing Symmetric Rhotrices: Properties and Applications
Keywords:
Symmetric Rhotrix, Eigenvalues, Diagonalization, Hessian Rhotrix, Positive Definiteness, Spectral TheoryAbstract
This study introduces symmetric rhotrices as an extension of symmetric matrices, establishing their foundational structure and defining key operations. We explore their properties, particularly in function analysis, through Hessian rhotrices, which aid in classifying critical points—except in cases where the determinant is zero. The research examines fundamental concepts such as positive definiteness, diagonalization, eigenvectors, and eigenvalues, using the Hessian Rhotrix of 4R , an even-dimensional rhotrix, as a case study. The spectral theorem is employed to determine diagonalizability, analyze vector transformations, and demonstrate vector
alignments within 4R . Additionally, we apply the second derivative test to classify critical points. This work contributes to the development of rhotrix algebra and its potential applications in fields such as machine learning, quantum mechanics, and optimization theory
References
Ajibade, O. A. (2003). The concept of Rhotrix in Mathematical Enrichment. International Journal of Mathematical Education in Science and Technology., 175-179. https://doi.org/10.1080/0020739021000053828
Aminu, A. (2010). The concept of Rhotrix Eigenvectors Eigenvalues Problem (REP). International Journal of Mathematics Education Science and Technology., 41, 98-105. https://doi.org/10.1080/00207390903189187
Atanassov, K. T., & Shannon, A. G. (1998). Matrix-Tertions and Matrix-Noitrets: Exercises in Matgematical Enrichment. International Journal of Mathematics Education Science and Technology. 29, 898-903.
Brown, D. E. (2014). Curvature, concavity, and eigenvalues of the Hessian matrix. Mathematical Analysis Review.
Isere, O. (2018). Even-dimensional rhotrices and their applications. Notes on Number Theory and Discrete Mathematics., 24(2), 125-133.https://doi.org/10.7546/nntdm.2018.24.2.125-133
Omolehin, J. (2006). Hessian Matrices and their Function Classification Properties. Applied Mathematics Journal.
Sani, B. (2004). An Alternative Method for Multiplication of Rhotrices. International Journal of Mathematical Education in Science and Technology, 35(5) 777-781. https://doi.org/10.1080/00207390410001716577
Usaini Salisu, A. M. (2014). On the Rhotrix eigenvalues and Eigenvectors. Afrika Matematika., 25, 223-235. http://dx.doi.org/10.1007/s13370-012-0103-9
Utoyo, T. O., Isere, A. O., & Ugbene, I. J. (2023). A New Multiplication approach with Applications in Differential and Integration of Even-Dimensional Hl-Rhotrices. Ambrose Alli University Journal of Physical and Applied Sciences, 3(1), 55-67.
Zhang, X. (2021). Symmetric Matrix Properties and Applications: A Guide. Built In. https://builtin.com/data-science/symmetric-matrix
