Abstract
Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a curve surrounding the spectrum. The logarithm function that can be found using this technique is essentially unique. To define a version of the logarithm with multiple values analogous to the one existing in the case of complex variables, we introduce a definition for the argument of a matrix, showing the possibility of finding equations similar to those of the scalar case. In the last section, numerical experiments performed by the first author, using the computer algebra program Mathematica©, confirm the effectiveness of this methodology. They include the logarithm of matrices of the fifth, sixth and seventh order.
Highlights
Academic Editor: Hari MohanMatrix functions are an important topic [1,2] due to their applications in many branches of applied mathematics, physics, statistics and engineering
We are able to represent the logarithm of a non-singular complex matrix using the Dunford–Taylor integral and apply Cauchy’s residue theorem if the matrix eigenvalues are known, or a simple circuit integral on a regular curve surrounding the spectrum, avoiding this knowledge
In [10], we have used the Dunford–Taylor integral in order to construct the roots of a non-singular complex matrix A according to the equation: A1/n =
Summary
Matrix functions are an important topic [1,2] due to their applications in many branches of applied mathematics, physics, statistics and engineering. We are able to represent the logarithm of a non-singular complex matrix using the Dunford–Taylor integral and apply Cauchy’s residue theorem if the matrix eigenvalues are known, or a simple circuit integral on a regular curve surrounding the spectrum, avoiding this knowledge. This can be done since it is always possible to find a ray r coming out from the origin not intersecting the spectrum of A and to consider a regular curve γ surrounding all the eigenvalues and such that r ∩ γ = ∅. Some numerical experiments were made by the first author using the computer algebra program Mathematica©, confirming the effectiveness of the proposed method
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
