Abstract
A method for the computation of the n th roots of a general complex-valued r × r non-singular matrix ? is presented. The proposed procedure is based on the Dunford–Taylor integral (also ascribed to Riesz–Fantappiè) and relies, only, on the knowledge of the invariants of the matrix, so circumventing the computation of the relevant eigenvalues. Several worked examples are illustrated to validate the developed algorithm in the case of higher order matrices.
Highlights
Complex-valued matrices are a natural extension of complex numbers, and matrix operations are well known to the reader [1] likely with the only exception of roots’ computation.The normal situation for a complex number is that the nth root always has n determinations.The equivalent situation for an r × r matrix A is that the nth root of A should have nr determinations.The problem arises in the case of matrices of a special type, for which the computation of roots is ill-posed in the sense of J.Hadamard, as they admit no roots or, an infinite number of those.In general, the problem of computing the nth roots of general complex-valued matrices has not received the necessary attention so far
The Dunford–Taylor integral [23] is an analogue of the Cauchy integral formula in function theory
The proposed procedure was based on the application of the Dunford–Taylor integral in combination with a suitable representation formula of the matrix resolvent
Summary
Complex-valued matrices are a natural extension of complex numbers, and matrix operations are well known to the reader [1] likely with the only exception of roots’ computation. The first method was presented in [11] and is based on the Cayley–Hamilton theorem in combination with the representation of non-singular matrix powers in terms of Chebyshev polynomials of the second kind [12,13] In this way, it is possible to express the roots of non-singular 2 × 2 or 3 × 3 complex-valued matrices by making use of pseudo-Chebyshev functions [14,15]. It is shown that the evaluation of the roots of a given non-singular matrix A can be performed only on the basis of the knowledge of the matrix invariants, which are the coefficients of the characteristic equation (or equivalently, the elementary symmetric functions of the eigenvalues), and the relevant spectral radius R, which can be estimated using Gershgorin’s theorem In this way, a numerical quadrature rule can be adopted to compute a contour integral extended to a circle centered at the origin and having a radius larger than R. The computer program Mathematica c is used
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