About square roots of matrices over an arbitrary field

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Koliada R. V., Melnyk O. M., Prokip V. M. № 2 (59) 56-64 Image Image

It is said that an matrix is the square root of an matrix over a field if and only if . The square roots of matrices are the main concepts for calculation of the polar decomposition of a matrix, solvability of differential equations and for construction of Markov financial model. It is evident that a matrix has a square root if and only if the matrix equation is solvable. It is known that square roots of complex numbers always exist, but square roots of matrices over the field of complex numbers may not exist. On the other hand, a non-zero matrix can have an infinite number of square roots. The problem of existence of square roots for matrices over the field of complex numbers is well studied. It may be noted that the problem of existence of square roots of matrices over an arbitrary field has been poorly researched.

In this article we study the problem of existence of square roots of matrices over an arbitrary field . Under certain restrictions we give necessary and sufficient conditions for existing the square root with given characteristic polynomial for a matrix over a field . It has been proved if the given square root exists, then it is uniquely defined by polynomial and we suggest a method of finding it. This allows us to give a complete solution to the problem of square roots of matrices over arbitrary field with relatively prime elementary divisors.

This research method allows us to specify the classes of matrices for which the number of square roots is finite.

Keywords: Square root of a matrix, rank of matrix, characteristic polynomial, li­near matrix equation.

doi: 10.32403/1998-6912-2019-2-59-56-64

  • 1. Gantmakher, F. R. (2004). Teoriia matritc. 5-e izd. Moskva : FIZMATLIT (in Russian).
  • 2. Gohberg, I., Lancaster, P., & Rodman, L. (1982). Matrix polynomials. New York : Academic Press (in English).
  • 3. Kazimirskyi, P. S. (1981). Rozklad matrychnykh mnohochleniv na mnozhnyky. Kyiv : Naukova dumka (in Ukrainian).
  • 4. Hoskins, W. D., & Walton, D. J. (1978). A faster method of computing square roots of a matrix: IEEE Transactions on Automatic Control, 23 (3), 494–495 (in English).
  • 5. Björck, A., & Hammarling S. (1982). A Schur method for the square root of a matrix: Linear Algebra and its Applications, 52/53, 127–140 (in English).
  • 6. Higham, N. J. (1986). Newton’s method for the matrix square root: Mathematics of Com­putation, 46 (174), 537–549 (in English).
  • 7. Sullivan, D. (1993). The square roots of 2×2 matrices: Math. Magazine, 66 (5), 314–316 (in English).
  • 8. Higham, N. J. (1997). Stable iterations for the matrix square root: Numerical Algorithms, 15 (2), 227–242 (in English).
  • 9. Hasan, M. A. (1997). A power method for computing square roots of complex matrices: Journal of Mathematical Analysis and Applications, 213 (2), 393–405 (in English).
  • 10. Johnson, C. R., Okubo, K., & Reams, R. (2001). Uniqueness of matrix square roots and an applications: Linear Algebra and its applications, 323 (1–3), 51–60 (in English).
  • 11. Johnson, C. R., & Okubo, K. (2002). Uniqueness of matrix square roots under a numerical range condition: Linear algebra and its applications, 341 (1–3), 195–199 (in English).
  • 12. Iannazzo, B. (2003). A note on computing the matrix square root. Calcolo: Springer, 40 (4), 273–283 (in English).
  • 13. Meini, B. (2004). The matrix square root from a new functional perspective: theoretical results and computational issues: SIAM Journal on Matrix Analysis and Applications, 26 (2), 362–376 (in English).
  • 14. Liu, Z., Zhang, Y., & Ralha, R. (2007). Computing the square roots of matrices with central symmetry: Applied Mathematics and Computation, 186 (1), 715–726 (in English).
  • 15. Zhang, Y., Li, W., Guo, D., & Ke, Zh. (2013). Different Zhang functions leading to different ZNN models illustrated via time-varying matrix square roots finding: Expert Systems with Applications, 40 (11), 4393–4403 (in English).
  • 16. Li, C. M., & Shen, S. Q. (2014). Newton’s Method for the Matrix Nonsingular Square Root: Journal of Applied Mathematics. Hindawi Pub. Corporation. Article ID 267042, 7 pages. doi: org/10.1155/2014/267042 (in English).
  • 17. Soleymani, F., Shateyi, S., & Khaksar, Haghani F. (2014). A numerical method for computing the principal square root of a matrix. Abstract and Applied Analysis: Hindawi Pub. Corporation. Article ID 525087, 7 pages. Retrieved from http://dx.doi.org/10.1155/2014/525087 (in English).
  • 18. Sadeghi, A. (2016). Approximating the principal matrix square root using some novel third-order iterative methods: Ain Shams Engineering Journal, 9 (4), 993–999 (in English).
  • 19. Nichols, J. (2016). A New Algorithm for Computing the Square Root of a Matrix. Thesis / Rochester Institute of Technology (in English).
  • 20. Del Moral, P., & Niclas, A. A. (2018). Taylor expansion of the square root matrix function: Journal of Mathematical Analysis and Applications, 465 (1), 259–266 (in English).
  • 21. Gawlik, E. S. (2019). Zolotarev Iterations for the Matrix Square Root: SIAM Journal on Matrix Analysis and Applications, 40 (2), 696–719 (in English).
  • 22. Kazimirskii, P. S., & Urbanovich, M. N. (1973). O razlozhenii matrichnogo dvuchlena na mnozhiteli: Ukrainskii matematicheskii zhurnal, 25 (4), 454–464 (in Russian).
  • 23. Melnyk, O. M., & Kolyada, R. V. (2017). On square roots of integer matrices. Book of Abstracts of 11th International Algebraic Conference in Ukraine. Kyiv (in English).
  • 24. Petrichkovich, V. M., & Prokip, V. M. (1986). O faktorizatcii mnogochlennykh matritc nad proizvolnym polem: Ukrainskii matematicheskii zhurnal, 38 (4), 478–483 (in Russian).
  • 25. Prokip, V. M. (1993). Pro yedynist unitalnoho dilnyka matrychnoho mnohochlena nad dovilnym polem: Ukrainskyi matematychnyi zhurnal, 45 (6), 803–808 (in Ukrainian).