# COMPUTING CAUCHY INTEGRAL THEOREM OF A MATRIX FUNCTION VIA SIMILARITY TRANSFORMATION AND BEHAVIOR OF HYPERGEOMETRIC MATRIX DENSITY FUNCTION

## Keywords:

65F60, 65F15, Secondary, 30E20, 28A25, primary 28A20, density of a matrix MSC 2020 (Mathematical Reviews and zbMath), Bessel matrix polynomial, hypergeometric function, Cauchy integral matrix function## Abstract

It is shown that Cauchy integral matrix function is Lebesgue measurable on the contour integral based on the Little Wood’s third formula for which the trapezoidal rule and Simpson composite method are applicable. As a follow up, it is demonstrated that Taylor series representation of Cauchy integral matrix function is commutable with similarity matrix transformation when Jordan canonical block along diagonal is taken into consideration. The spectrum of the diagonalizable matrix A is computed using the Givens

orthogonal matrix plane rotation. As an extension of ideas; a measure of effectiveness on the use of SVD in the computation process is emphasized and fully utilized which leads to demonstration with the exponential of a matrix function as an example. Further analytical reasoning on performance of Cauchy integral theorem for the matrix functional calculus leads to the method of Residue theorem. The

density of a matrix function is calculated based on the hypergeometric series taking into consideration the behavior of gamma function.

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## References

Takihira,S., Ohashi,A. , Sogabe,T,Usuda,T.S(2003). Quantum algorithm for matrix functions by Cauchy’s integral theorem formula. Quantum Information and Computation, Vol. 0, No. 0, pp. 00-00. rXiv:2106.08075v1[quant- ph] June,2021.

Higham, N.J (2008). Functions of Matrices : Theory and Computation, SIAM, Philadelphia.

Kaplan,W (1959). Advanced Calculus. Addison-Wesley Publishing Company,Inc. U.S.A.

Uwamusi,S.E (2014). Measure Theory and Integration. Lecture Notes ,Department of Mathematics, Kogi State University, Anyigba ,Kogi State,Nigeria.

Taylor, M.E(2006). Measure theory and integration. Graduate Studies in Mathematics, Vol. 76, AMS. ISBN- 10:8218-4180-7

Rudin,W (1976).Principles of Mathematical Analysis (3 rd edition) McGraw-Hill, Kogakusha, Internationa Student Edition.

Teleman,C(2003). Rieman Surfaces. Lent Lecture Notes. https://math.Berkeley.edu

Jarlebring, E (2016). Lecture Notes in numerical algebra KTH, Autumn 2015, version 2016.

Neumaier, A (2001). Introduction to Numerical analysis. Academic Press, New York .

Bjorck, A. (2009) . Numerical methods in Scientific Computing Vol. 2, SIAM, Philadelphia.

Uwamusi,S.E (2021). The Gamma and Beta Matrix Functions and Other Applications. Unilag Journal of Mathematics and Applications, Volume 1, Issue 2,pp. 186-207. ISSN: 2805-3966. URL:https://lagjma.edu.ng

Stephen Trans. Of NAMP Zabarankin,M (2012). Cauchy integral formula for generalized analytic functions in hydrodynamics. Proceedings

of the Royal Society, 468, pp 3745-3764.

Rainville, E.D(1960). Special functions, Chelseas, New York. Hannah,J.P(2013) . Identities for the Gamma and Hypergeometric functions: an overview from Euler to the present. M.Sc. Thesis, School of Mathematics, University of the Witwatersrand Johasnesburg, South Africa.

Zhao,T.H., Wang,M.K.,Zhang,W,and Chu, Y.M (2018)., Quadratic transformation inequalities for Gaussian hypergeometric Function. Journal of Inequalities and Applications.:251.https:// doi.org/10.1186/S136650-018-1848

Jodar,L. and Cortis,J.C (1998). Some properties of Gamma and Beta matrix functions, Appl. Math. Lett. 11(1) , 89-93.

Uwamusi,S.E (2017). Extracting p-th root of a matrix with positive eigenvalues via Newton and Halley’s methods. Ilorin Journal of Science, Vol. 4 (1), pp. 1-16. ISSN:2408-4840.

Nagar,K. D., Joshi, L.,Gupta, A.(2012). Matrix variate Pareto distribution of the second kind. International Scholarly Research Network, ISRN Probability and Statistics Vol. 2012, Article ID 789273, 20 pages. Doi:10.5402/2012/789273

Uwamusi,S.E, and Otunta,F.O (2002). Computation of eigenvalues of Hermitian matrix via Givens plane rotation. Nigerian Journal of Applied Science Vol. 20 ,pp.101-106.

Uwamusi,S.E (2005). A Class of algorithms for Zeros of polynomials. Parkistan Journal for Scientific and Industrial Research. 48(3) ,pp. 149-153.

Frerix, T. and Bruna, J (2019). Approximating Orthogonal matrices with effective Givens factorization. Proceedings of the 36th International Conference of Machine Learning, Long Beach California.

Wilkinson,J.H (1965).The Algebraic Eigenvalue Problem. Oxford University Press, Ely House, London, UK. Pp. 282-307

Golub, G and Vanloan, F(1983). Matrix computations. The Johns Hopkins University Press, Baltimore Maryland.

Horn,R.A and Johnson,C.R(1993). Matrix Analysis. Cambridge University Press.

Chang, X and Li, R. (2011). Multiplicative perturbation analysis for QR Factorizations. Mathematics Preprint Series. Technical Report 2011-01, 18 pages. https://w.w.w.uta/math/preprint/

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