# 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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