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On the history of the Bézoutian and the resultant matrix. (English) Zbl 0711.15028

This exciting story starts with two algorithms given by Euler (1748) to construct, in the ideal generated by F(x,y) and G(x,y), a polynomial independent of Y. Bézout (1764) described a reduction method which leads to a certain system of linear equations. It was Cayley (1857) who observed that the matrix of this system may be identified with the matrix known today as the Bézoutian, i.e. the matrix whose generating function is the polynomial \((f(x)g(y)-g(x)f(y))/(x-y)\).
Reviewer: V.Pták

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A06 Linear equations (linear algebraic aspects)
15-03 History of linear algebra
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References:

[1] Bezout, É., Recherches sur le degré des équations résultantes de l’évanouissement des inconnues, et sur les moyens qu’il convient d’employer pour trouver ces équations, Mém. Acad. Roy. Sci. Paris, 288-338 (1764)
[2] Cayley, A., Note sur la méthode d’élimination de Bezout, J. Reine Angew. Math., 53, 366-367 (1857) · ERAM 053.1409cj
[3] Dieudonné, J., Abrégé d’Histoire des Mathématiques 1700-1900 (1978), Hermann: Hermann Paris, Tome I · Zbl 0656.01001
[4] Euler, L., Introductio in Analysin Infinitorum (1748), Tom. 2, Lausanne · Zbl 0096.00303
[5] Jacobi, C. G.J., De eliminatione variabilis e duabus aequationibus algebraicis, J. Reine Angew. Math., 15, 101-124 (1836) · ERAM 015.0531cj
[6] Kailath, T., Linear Systems (1980), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0458.93025
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