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Logarithm of a Non-Singular Complex Matrix via the Dunford-Taylor Integral

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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… Click to show full 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.

Keywords: dunford taylor; non singular; taylor integral; matrix; complex matrix; singular complex

Journal Title: Axioms
Year Published: 2022

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