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Title: An identity between the determinant and the permanent of Hessenberg-type matrices (English)
Author: da Fonseca, Carlos Martins
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 61
Issue: 4
Year: 2011
Pages: 917-921
Summary lang: English
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Category: math
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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. (English)
Keyword: determinant
Keyword: permanent
Keyword: Hessenberg matrices
Keyword: graphs
Keyword: trees
MSC: 05C50
MSC: 15A15
idZBL: Zbl 1249.15011
idMR: MR2886247
DOI: 10.1007/s10587-011-0059-1
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Date available: 2011-12-16T15:37:30Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/141797
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Reference: [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. Zbl 1100.68037, MR 2275715
Reference: [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, 10.1090/dimacs/004/09
Reference: [3] Botta, P.: On the conversion of the determinant into the permanent.Can. Math. Bull. 11 (1968), 31-34. Zbl 0159.32201, MR 0230734, 10.4153/CMB-1968-004-6
Reference: [4] Elsner, L.: A note on generalized Hessenberg matrices.Linear Algebra Appl. 409 (2005), 147-152. Zbl 1082.15038, MR 2170273
Reference: [5] Fiedler, M., Vavřín, Z.: Generalized Hessenberg matrices.Linear Algebra Appl. 380 (2004), 95-105. MR 2038742, 10.1016/S0024-3795(03)00555-X
Reference: [6] Gibson, P. M.: An identity between permanents and determinants.Am. Math. Mon. 76 (1969), 270-271. Zbl 0174.31505, MR 0241439, 10.2307/2316368
Reference: [7] Gibson, P. M.: Conversion of the permanent into the determinant.Proc. Am. Math. Soc. 27 (1971), 471-476. Zbl 0194.06002, MR 0279110, 10.1090/S0002-9939-1971-0279110-X
Reference: [8] Hwang, S.-G., Kim, S.-J., Song, S.-Z.: On convertible complex matrices.Linear Algebra Appl. 233 (1996), 167-173. Zbl 0856.15008, MR 1368080
Reference: [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. Zbl 1042.15007, MR 1976858, 10.1080/0308108031000114686
Reference: [10] Kräuter, A. R., Seifter, N.: On convertible $(0,1)$-matrices.Linear Multilinear Algebra 13 (1983), 311-322. Zbl 0522.15003, MR 0704780, 10.1080/03081088308817530
Reference: [11] Lim, M. H.: A note on the relation between the determinant and the permanent.Linear Multilinear Algebra 7 (1979), 45-47. Zbl 0403.15006, MR 0529883
Reference: [12] Marcus, M., Minc, H.: On the relation between the determinant and the permanent.Ill. J. Math. 5 (1961), 376-381. Zbl 0104.00904, MR 0147488, 10.1215/ijm/1255630882
Reference: [13] McCuaig, W.: Pólya's permanent problem.Electron. J. Comb. 11 (2004), Research paper R79. Zbl 1062.05066, MR 2114183
Reference: [14] Pólya, G.: Aufgabe 424.Arch. Math. Phys. Ser. 3 20 (1913), 271.
Reference: [15] Reich, S.: Another solution of an old problem of Pólya.Am. Math. Mon. 78 (1971), 649-650. Zbl 0224.15006, MR 1536369, 10.2307/2316574
Reference: [16] Szegö, G.: Lösung zu Aufgabe 424.Arch. Math. Phys. Ser. 3 21 (1913), 291-292.
Reference: [17] Tarakanov, V. E., Zatorskiĭ, R. A.: A relationship between determinants and permanents.Math. Notes 85 (2009), 267-273. MR 2548008, 10.1134/S0001434609010301
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