In this paper we extend the results of an earlier note [1].

Definition. Let E be an extension field of the rationals. A vector v = (b1, …, bn) in En is algebraic if each coordinate bi is algebraic over the rationals. A linear transformation T: En → En is algebraic if T(v) is an algebraic vector for every algebraic vector v.

Definition. The degree of an algebraic linear transformation T, denoted by deg T, is the minimum of [K:Q] taken over all finite algebraic extensions K of the rationals Q such that T: Kn → Kn.

It is well known that if A is an n × n complex matrix and AT is its transpose, then there is an invertible n x n complex matrix S such that AT = S-1AS. In this note we wish to point out another simple relation between A and AT.

If A is an n × n complex matrix and AT is its transpose then there are unitary n × n complex matrices U and V such that AT = UAV.

Dans un récent article [2, p. 379], R. M. Robinson a émis une conjecture intéressante sur le diamètre transfini des ensembles compacts et symétriques du plan. La conjecture est la suivante:

Soit E un ensemble compact de points du plan, symétrique par rapport á l'axe réel R. Pour n ≥ 2, choisissons n points z1, …, zn de E et posons xi = zi,   pour i=1, 2,…, n.

Soient: