Abstract
In this short note we provide an extension of the notion of Hessenberg matrix and observe an identity between the determinant and the permanent of such matrices. The celebrated identity due to Gibson involving Hessenberg matrices is consequently generalized.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Agrawal: Determinant versus permanent. Proceedings of the international congress of mathematicians, Madrid, Spain, August 22–30, 2006. European Mathematical Society, Zürich, 2006, pp. 985–997.
R.A. Brualdi, B. L. Shader: On sign-nonsingular matrices and the conversion of the permanent into the determinant. Applied geometry discrete mathematics. DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4 (1991), 117–134.
P. Botta: On the conversion of the determinant into the permanent. Can. Math. Bull. 11 (1968), 31–34.
L. Elsner: A note on generalized Hessenberg matrices. Linear Algebra Appl. 409 (2005), 147–152.
M. Fiedler, Z. Vavřín: Generalized Hessenberg matrices. Linear Algebra Appl. 380 (2004), 95–105.
P.M. Gibson: An identity between permanents and determinants. Am. Math. Mon. 76 (1969), 270–271.
P.M. Gibson: Conversion of the permanent into the determinant. Proc. Am. Math. Soc. 27 (1971), 471–476.
S.-G. Hwang, S.-J. Kim, S.-Z. Song: On convertible complex matrices. Linear Algebra Appl. 233 (1996), 167–173.
M.P. Coelho, M.A. Duffner: On the relation between the determinant and the permanent on symmetric matrices. Linear Multilinear Algebra 51 (2002), 127–136.
A.R. Krauter, N. Seifter: On convertible (0, 1)-matrices. Linear Multilinear Algebra 13 (1983), 311–322.
M.H. Lim: A note on the relation between the determinant and the permanent. Linear Multilinear Algebra 7 (1979), 45–47.
M. Marcus, H. Minc: On the relation between the determinant and the permanent. Ill. J. Math. 5 (1961), 376–381.
W. McCuaig: Pólya’s permanent problem. Electron. J. Comb. 11 (2004). Research paper R79.
G. Polya: Aufgabe 424. Arch. Math. Phys. Ser. 3 20 (1913), 271.
S. Reich: Another solution of an old problem of Pólya. Am. Math. Mon. 78 (1971), 649–650.
G. Szego: Lösung zu Aufgabe 424. Arch. Math. Phys. Ser. 3 21 (1913), 291–292.
V.E. Tarakanov, R.A. Zatorskiı: A relationship between determinants and permanents. Math. Notes 85 (2009), 267–273.
Author information
Authors and Affiliations
Corresponding author
Additional information
This note is supported by CMUC — Centro de Matemática da Universidade de Coimbra.
Rights and permissions
About this article
Cite this article
da Fonseca, C.M. An identity between the determinant and the permanent of Hessenberg-type matrices. Czech Math J 61, 917–921 (2011). https://doi.org/10.1007/s10587-011-0059-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-011-0059-1