Research article

On numerical/non-numerical algebra: Semi-tensor product method

  • Received: 10 February 2021 Accepted: 13 March 2021 Published: 15 March 2021
  • A kind of algebra, called numerical algebra, is proposed and investigated. As its opponent, non-numerical algebra is also defined. The numeralization and dis-numeralization, which convert non-numerical algebra to numerical algebra and vise versa, are considered. Product structure matrix (PSM) of a finite dimensional algebra is constructed. Using PSM, some fundamental properties of finite dimensional algebras are obtained. Then a necessary and sufficient condition for a numerical algebra to be a field is presented. Finally, the invertibility of Segre (commutative) quaternion and some related properties of matrices over Segre quaternion are investigated.

    Citation: Daizhan Cheng, Ying Li, Jun-e Feng, Jianli Zhao. On numerical/non-numerical algebra: Semi-tensor product method[J]. Mathematical Modelling and Control, 2021, 1(1): 1-11. doi: 10.3934/mmc.2021001

    Related Papers:

  • A kind of algebra, called numerical algebra, is proposed and investigated. As its opponent, non-numerical algebra is also defined. The numeralization and dis-numeralization, which convert non-numerical algebra to numerical algebra and vise versa, are considered. Product structure matrix (PSM) of a finite dimensional algebra is constructed. Using PSM, some fundamental properties of finite dimensional algebras are obtained. Then a necessary and sufficient condition for a numerical algebra to be a field is presented. Finally, the invertibility of Segre (commutative) quaternion and some related properties of matrices over Segre quaternion are investigated.



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    [1] V. Abramov, Noncommutative galois extension and graded q-differential algebra, Adv. Appl. Clifford Alg., 26 (2016), 1–11. doi: 10.1007/s00006-015-0599-9
    [2] W. Boothby, An Introduction to Differential Manifold and Riemannian Geometry, Orlando: Academic Press, 1986.
    [3] F. Catoni, R. Cannata, P. Zampetti, An introduction to commutative quaternions, Adv. Appl. Clifford Alg., 16 (2005), 1–28.
    [4] D. Cheng, Some applications of semi-tensor product of matrices in algebra, Comp. Math. Appl., 52 (2005), 1045–1066.
    [5] D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks–A Semi-tensor Product Approach, London: Springer, 2011.
    [6] D. Cheng, H. Qi, Y. Zhao, An Introduction to Semi-tensor Product of Matrices and Its Applications, Singapore: World Scientific, 2012.
    [7] D. Cheng, H. Qi, Y. Zhao, Analysis and control of general logical networks–An algebraic approach, Annu. Rev. Control, 36 (2012), 11–25. doi: 10.1016/j.arcontrol.2012.03.002
    [8] D. Cheng, On equivalence of matrices, Asian J. Math., 23 (2019), 257–348.
    [9] D. Cheng, From Dimensional-Free Matrix Theory to Cross-Dimensional Dynamic Systems, London: Elsevier, 2019.
    [10] C. Clarke, Elementary General Relativity, New York: John Wiley & Sons, 1979.
    [11] S. Fu, D. Cheng, J. Feng, J. Zhao, Matrix expression of finite Boolean-type algebras, Applied Math. Comput., 395 (2021), 125880.
    [12] T. Hungerford, Algebra, Springer-Verlag, 1974.
    [13] X. Kong, T. Tong, Ral representation and inverse matrix method of the mixed tyoe comutative quaternion matrix, J. Northeas Normal Univ. (Natural Sci. Ed.)., 51 (2019), 15–20 (in Chinese).
    [14] S. Lang, Algebra, Revised Third Edition, New York: Springer-Verlag, 2002.
    [15] W. Li, Lecture on History of Mathematics, Beijing: Higher Edication Press, 1999 (in Chinese).
    [16] P. Morandi, Field and Galois Theory, New York: Springer-Verlag, 1996.
    [17] S. Pei, J. Chang, J. Ding, Commutative reduced biquaternions and their fourier transform for sigmal and image processing applications, IEEE Trans. Signal Proc., 52 (2004), 2012–2031. doi: 10.1109/TSP.2004.828901
    [18] D. Pinotsis, Segre quaternions, spectral analysis and a four-dimensional laplace equation. In Progress in Analysis and its Applications, M. Ruzhansky and J. Wirth, eds, World Scientific, Singapore, 240-246 (2010).
    [19] J. Rooncy, On the three types of compex number and planar transformation, Environ. Plann. B., 5 (1978), 89–99. doi: 10.1068/b050089
    [20] C. Segre, The real representations of complex elements and extension to bicomplex systems, Math. Ann., 40 (1892), 413–467. doi: 10.1007/BF01443559
    [21] A. Trovon, Q. Suzuki, Noncommutative galois extensions and ternary clifford analysis, Adv. Appl. Clifford Alg., 27 (2017), 59–70.. doi: 10.1007/s00006-015-0565-6
    [22] J. Yaglom, Complex Numbers in Geometry, New York: Academic Press, 1968.
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