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Explicit Formulas for the Matrix Logarithm and the Principal Matrix Logarithm

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Abstract

We present new closed-form formulas for the matrix logarithm. Our method is direct and elementary, and it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Using these results, an elegant explicit formula for the principal matrix logarithm can be obtained. Several special cases and examples are stated to illustrate the method presented in this paper.

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References

  1. C. D. Ahlbrandt and J. Ridenhour, “Floquet theory for time scales and Putzer representations of matrix logarithms,” J. Differ. Equ. Appl. 9, 77–92 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  2. N. J. Higham, Functions of Matrices: Theory and Applications (SIAM-Philadelphia, 2008), Vol. 18.

    Book  MATH  Google Scholar 

  3. S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to specified accuracy,” SIAM. J. Matrix. Anal. Appl. 22, 1112–1125 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  4. C. S. Kenney and A. J. Laub, “Condition estimates for matrix functions,” SIAM. J. Matrix. Anal. Appl. 10, 191–209 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. R. Cardoso, “An explicit formula for the matrix logarithm,” S. Afr. Optom. 64, 80–83 (2005).

    Google Scholar 

  6. J. A. Marrero, R. B. Taher, and M. Rachidi, “On explicit formulas for the principal matrix logarithm,” Appl. Math. Comput. 220, 142–148 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. H. Al-Mohy and N. J. Higham, “Improved inverse scaling and squaring algorithms for the matrix logarithm,” SIAM. J. Sci. Comput. 34, C153–C169 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Miyajima, “Verified computation for the matrix principal logarithm,” Linear Algebra Appl. 569, 38–61 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Mouçouf, An Explicit Expression for the Minimal Polynomial of the Kronecker Product of Matrices. Explicit Formulas for Matrix Logarithm and Matrix Exponential, arXiv: 2010.11873v3

    Google Scholar 

  10. M. Mouçouf and S. Zriaa, “A new approach for computing the inverse of confluent Vandermonde matrices via Taylor’s expansion,” Linear Multilinear Algebra 70, 5973–5986 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Mouçouf and S. Zriaa, “Explicit formulas for the matrix exponential,” Boletim da Sociedade Paranaense de Matemática. 41, 1–14 (2023).

    Article  MathSciNet  MATH  Google Scholar 

  12. H. W. Gould, “Euler’s formula for the nth differences of powers,” Amer. Math. Monthly 85, 450–467 (1978).

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors wish to express gratitude to the referee for carefully reading the manuscript.

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Authors and Affiliations

  1. Faculty of Science, Department of Mathematics, University Chouaib Doukkali, Eljadida, 24000, Morocco

    M. Mouçouf & S. Zriaa

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  1. M. Mouçouf
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  2. S. Zriaa
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Correspondence to M. Mouçouf.

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Mouçouf, M., Zriaa, S. Explicit Formulas for the Matrix Logarithm and the Principal Matrix Logarithm. Math Notes 113, 406–413 (2023). https://doi.org/10.1134/S0001434623030094

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  • DOI: https://doi.org/10.1134/S0001434623030094

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