Authors : Langote Ulhas Baban; Dr. Mulay Prashant P.

Volume/Issue : Volume 10 - 2025, Issue 1 - January

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Google Scholar : https://tinyurl.com/3z7aez79

Scribd : https://tinyurl.com/nu3y5bwj

DOI : https://doi.org/10.5281/zenodo.14717024

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Abstract : To study of root of matrix it is necessary to know about properties and some results of matrix. We try to prove such known and unknown results and analyze stability property and convergence of proof.

Keywords : M3 (R) is a set of square matrices of order 3 with all real entries. b) B = A 1/p is matrix A 1/p root of A. c) t = Trace of matrix B. Whereas tr(A) is trace of matrix A. d) ∆ = Determinant of B. Whereas |A| = det(A) is determinant of A. e) adj(B) = Adjoint of matrix B. f) ∝= tr(adj(B)) = Trace of adjoint of matrix B. g) In is identity matrix of order n. Though we use I as identity matrix of order 3.

References :

1. P. Benner, R. Byers, V. Mehrmann and H. Xu, A unified deflating subspace approach for classes of polynomial and rational matrix equations, Preprint SFB393/00-05, ZentrumfürTechnomathematik, Universität Bremen, Bremen, Germany (January 2000).
2. M.A. Hasan, J.A.K. Hasan and L. Scharenroich, New integral representations and algorithms for computing nth roots and the matrix sector function of nonsingular complex matrices, in: Proc. of the 39th IEEE Conf. on Decision and Control, Sydney, Australia (2000) pp. 4247–4252.
3. N.J. Higham, Newton’s method for the matrix square root, Math. Comp. 46(174) (1986) 537–549.
4. W.D. Hoskins and D.J. Walton, A faster, more stable method for computing the pth roots of positive definite matrices, Linear Algebra Appl. 26 (1979) 139–163.
5. L.-S. Shieh, Y.T. Tsay and R.E. Yates, Computation of the principal nth roots of complex matrices, IEEE Trans. Automat. Control 30(6) (1985) 606–608.
6. M.I. Smith, ASchur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl. 24(4) (2003) 971–989.
7. J.S.H. Tsai, L.S. Shieh and R.E. Yates, Fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function, Comput. Math. Appl. 15(11) (1988) 903–913.
8. Y.T. Tsay, L.S. Shieh and J.S.H. Tsai, A fast method for computing the principal nth roots of complex matrices, Linear Algebra Appl. 76 (1986) 205–221).