The Effect of the Additive Row Operation on the Permanent

Year 2023, Issue: 42, 8 - 13, 31.03.2023

Ahmet Zahid Küçük Abdullah Talha Sözer

https://doi.org/10.53570/jnt.1178990

Abstract

The permanent function is not as stable as the determinant function under the elementary row operations. For example, adding a non-zero scalar multiple of a row to another row does not change the determinant of a matrix, but this operation changes its permanent. In this article, the variation in the permanent by applying the operation, which adds a scalar multiple of a row to another row, is examined. The relationship between the permanent of the matrix to which this operation is applied and the permanent of the initial matrix is given by a theorem. Finally, the paper inquires the need for further research.

Keywords

Permanent, Gaussian elimination, elementary row operations, Laplace expansion

References

• H. Minc, M. Marcus, Permanents, Vol. 6 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1984.
• R. A. Brualdi, Henryk Minc, Permanents, Bulletin (New Series) of the American Mathematical Society 1 (6) (1979) 965–973.
• V. Kocharovsky, V. Kocharovsky, V. Martyanov, S. Tarasov, Exact Recursive Calculation of Circulant Permanents: A Band of Different Diagonals Inside a Uniform Matrix, Entropy 23 (11) (2021) 1423 17 pages.
• R. B. Bapat, Recent Developments and Open Problems in the Theory of Permanents, The Mathematics Student 76 (1-4) (2007) 55–69.
• R. A. Brualdi, D. Cvetkovic, A Combinatorial Approach to Matrix Theory and Its Applications, Chapman and Hall/CRC, Boca Raton, 2008.
• M. Fiedler, F. J. Hall, A Note on Permanents and Generalized Complementary Basic Matrices, Linear Algebra and Its Applications 436 (9) (2012) 3553–3561.
• M. Fiedler, F. J. Hall, M. Stroev, Permanents, Determinants, and Generalized Complementary Basic Matrices, Operators and Matrices 8 (4) (2014) 1041–1051.
• M. Stroev, Some Results on Generalized Complementary Basic Matrices and Dense Alternating Sign Matrices, Doctoral Dissertation Georgia State University (2016) Atlanta.
• X. Niu, S. Su, J. Zheng, A New Fast Computation of a Permanent, in: Institute of Physics (IOP) Conference Series: Materials Science and Engineering, Vol. 790, 2020 012057 7 pages.