The Application of Minimum Polynomial

Wenjuan Wang

doi:https://doi.org/10.14445/22315373/IJMTT-V68I10P503

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 10 | Year 2022 | Article Id. IJMTT-V68I10P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I10P503

The Application of Minimum Polynomial

Wenjuan Wang

Received	Revised	Accepted	Published
28 Aug 2022	06 Oct 2022	17 Oct 2022	28 Oct 2022

Citation :

Wenjuan Wang, "The Application of Minimum Polynomial," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 10, pp. 15-19, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I10P503

Abstract

Matrix theory plays an important role in algebra research and linear control, and the minimum polynomial plays an important role in matrix. Therefore, it is necessary to study the application of minimal polynomials. In this paper, six applications of minimum polynomial include solving matrix function and its inverse function by using minimum polynomial, determining singularity of matrix polynomial, solving matrix equation, determining whether matrix can be diagonalized, solving basis and dimension of linear space, and solving differential equation with constant coefficient.

Keywords

The matrix function and its inverse, The singularity of matrix polynomials, Diagonalizable, The basis and dimension.

References

[1] L.H. Zhang, and L.L. Wu, "The Application of Carley-Hamilton Theorem," Journal of Dezhou College, vol. 34, no. 2, pp. 1-8, 2018.
[2] L.H. Li, "Applications of the Hamilton-Cayley Theorem," Journal of Shanghai Electric Power University, vol. 24, no. 2, pp. 192-194, 2008.
[3] G.J. Li, "The Proof and Study of Hamilton-Cayley Theorem," Practice and Understanding of Mathematics, vol. 10, no. 1, pp. 46-47, 2003.
[4] C.D Jiao, "Study on Necessary and Sufficient Conditions for the Equality of Characteristic Polynomial and Minimum Polynomial," Journal of Jingdezhen College, vol. 35, no.153, pp.113-116, 2020.
[5] J.M. Yang, and J. Cao, "Elementary Transformation Method for Finding the Minimum Polynomial of Matrix," Practice and Understanding of Mathematics, vol. 34, no. 10, pp. 1-3, 2004.
[6] Z.X. Li, "The Method of Finding the Least Polynomial of Matrix," Journal of Shanxi Datong University: Natural Science Edition, vol. 34, no. 6, pp. 1-3, 2018.
[7] Y.U. Bo, J. Zhang, and Y.Y. XU, "The RCH Method for Computing Minimal Polynomials of Polynomial Matrices," Systems Science and Complexity, vol. 28, no. 1, pp. 190-209, 2015.
[8] D. Jie ,“The Minimum Polynomial of the Symmetric Array and Its,Application ”,Journal of Mathematics, vol. 31, no. 6, 2010.
[9] Z.L. Jiang and S.Y. Liu, "An Algorithm for Finding Minimal Polynomials of Squamous Cyclic Factor Matrices," Applied Mathematics, vol. 17, no. 1, pp. 1-6, 2004.
[10] Z. Bartosiewicz, "Minimal Polynomial Realizations," Journal of Applied Mathematics, vol. 16, no. 3, pp. 295-301, 1993.
[11] K.W. Huang, "Find the Basis Solution Matrix of Linear Differential Equations with Minimal Polynomial," Journal of Shaoxing University of Arts and Sciences: Natural Science Edition, vol. 26, no. 1, pp. 1-4, 2006.
[12] D.P. Hu, "Minimal Polynomial Solution of Matrix Equation AX - XB=C," Journal of Applied Mathematics, vol. 16, no. 3, pp. 295-301, 1993.
[13] T.M. Hoang, and X. Thierauf, "The Complexity of the Characteristic and the Minimal Polynomial," Theoretical Computer Science, vol. 1-3, no. 295, pp. 205-222, 2003.
[14] Z.X. Li ,“The Method of Finding the Minimum Polynomial of Matrix ”,Journal of Shanxi Datong University: Natural Science Edition, vol. 34, no. 6, pp. 1-3, 2018.
[15] B.L. Xia ,“Properties and Solutions of Square,Matrix,Minimal,Polynomials ”,Advanced Mathematical Research, vol. 15, no. 3, pp. 34- 39, 2003.
[16] J. H. Wu ,“The Solution and Application of Matrix,Minimal,Polynomial ”,Journal of Hanshan Normal University, vol. 10, no. 6, pp. 25- 30, 2010.
[17] L.H. Wang, and J.P. Wang ,“Properties of Minimal Polynomials and Their Applications ” Journal of Henan Institute of Education: Natural Science Edition, vol. 13, no.2, pp. 1-2, 2004.
[18] F.C. Feng ,“The Solution of the Minimum Polynomial of Matrix and Its,Application ”,Journal of Ningxia Normal University, vol. 38, no.6, pp. 1-5, 2017.
[19] H.Q. Ma, and J.J. Song ,“The Minimum Polynomial of Square Matrix and Its,Application ”,Journal of Nanyang Institute of Technology, vol. 3, no.16 (04), pp. 122-128, 2011.
[20] J. An ,“A Variety of Proofs and Applications of an Important Property of Minimal Polynomials ”,Advanced Mathematical Research, vol.23, no.1, pp.1-4, 2020.
[21] Z.Y. Wang, and H.R. Chen, "The Application of Matrix Minimal Polynomial in Differential Equations," Journal of Tongling University, 2004.
[22] Bartosiewicz, Z.,“Minimal Polynomial,Realizations ”,Mathematics of Control Signals & Systems, vol. 3, no. 1, pp. 227-237, 1998.
[23] Yu. B, Zhang. J, and Xu. Y ,“The RCH Method for Computing Minimal Polynomials of Polynomial,Matrices ”,System Science and Complex, vol. 28, no. 1, pp. 190-209, 2015.
[24] Aminu. A ,“Minimal Polynomial of A,Rhotrix ”,Afrika Matematika, vol. 26, no. 2, pp. 257-264, 2015.
[25] L. Huang, and N.C. Yu, "Minimal Polynomial Matrices and Linear Multivariable Systems (ⅱ)," Applied Mathematics and Mechanics, vol. 10, no. 6, pp. 905-922, 1985.