Previous |  Up |  Next

Article

Keywords:
determinant; permanent; Hessenberg matrices; graphs; trees
Summary:
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.
References:
[1] Agrawal, M.: Determinant versus permanent. Proceedings of the international congress of mathematicians, Madrid, Spain, August 22-30, 2006 European Mathematical Society Zürich (2006), 985-997. MR 2275715 | Zbl 1100.68037
[2] Brualdi, R. A., Shader, B. L.: 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. MR 1116343
[3] Botta, P.: On the conversion of the determinant into the permanent. Can. Math. Bull. 11 (1968), 31-34. DOI 10.4153/CMB-1968-004-6 | MR 0230734 | Zbl 0159.32201
[4] Elsner, L.: A note on generalized Hessenberg matrices. Linear Algebra Appl. 409 (2005), 147-152. MR 2170273 | Zbl 1082.15038
[5] Fiedler, M., Vavřín, Z.: Generalized Hessenberg matrices. Linear Algebra Appl. 380 (2004), 95-105. DOI 10.1016/S0024-3795(03)00555-X | MR 2038742
[6] Gibson, P. M.: An identity between permanents and determinants. Am. Math. Mon. 76 (1969), 270-271. DOI 10.2307/2316368 | MR 0241439 | Zbl 0174.31505
[7] Gibson, P. M.: Conversion of the permanent into the determinant. Proc. Am. Math. Soc. 27 (1971), 471-476. DOI 10.1090/S0002-9939-1971-0279110-X | MR 0279110 | Zbl 0194.06002
[8] Hwang, S.-G., Kim, S.-J., Song, S.-Z.: On convertible complex matrices. Linear Algebra Appl. 233 (1996), 167-173. MR 1368080 | Zbl 0856.15008
[9] Coelho, M. P., Duffner, M. A.: On the relation between the determinant and the permanent on symmetric matrices. Linear Multilinear Algebra 51 (2002), 127-136. DOI 10.1080/0308108031000114686 | MR 1976858 | Zbl 1042.15007
[10] Kräuter, A. R., Seifter, N.: On convertible $(0,1)$-matrices. Linear Multilinear Algebra 13 (1983), 311-322. DOI 10.1080/03081088308817530 | MR 0704780 | Zbl 0522.15003
[11] Lim, M. H.: A note on the relation between the determinant and the permanent. Linear Multilinear Algebra 7 (1979), 45-47. MR 0529883 | Zbl 0403.15006
[12] Marcus, M., Minc, H.: On the relation between the determinant and the permanent. Ill. J. Math. 5 (1961), 376-381. MR 0147488 | Zbl 0104.00904
[13] McCuaig, W.: Pólya's permanent problem. Electron. J. Comb. 11 (2004), Research paper R79. MR 2114183 | Zbl 1062.05066
[14] Pólya, G.: Aufgabe 424. Arch. Math. Phys. Ser. 3 20 (1913), 271.
[15] Reich, S.: Another solution of an old problem of Pólya. Am. Math. Mon. 78 (1971), 649-650. DOI 10.2307/2316574 | MR 1536369 | Zbl 0224.15006
[16] Szegö, G.: Lösung zu Aufgabe 424. Arch. Math. Phys. Ser. 3 21 (1913), 291-292.
[17] Tarakanov, V. E., Zatorskiĭ, R. A.: A relationship between determinants and permanents. Math. Notes 85 (2009), 267-273. DOI 10.1134/S0001434609010301 | MR 2548008
Partner of
EuDML logo