The Equivalence of Non-Singular Pencils of Hermitian Matrices in an Arbitrary Field
John H. Williamson · American Journal of Mathematics · 1935
The problem of the equivalence of two non-singular pencils of real symmetric matrices in the real field was first solved by Mluth.1 More recently Trott,2 Wegner,3 Ingraham 4 and Turnbull 5 have solved the similar problem for two Hermitian matrices under conjunctive transformations in the complex field. The notation used by Trott was such, that he was able to discuss the Hermitian case and at the same time the real symmetric case. In this paper we show how Trott's method may be extended to the similar problem of the equivalence of two non-singular pencils of Hermitian (or symmetric) matrices with respect to a general commutative field K. Incidentally, as is often the case with a generalization, we show why the results in the case of the complex field (or real field) are comparatively simple. We prove that a necessary and sufficient condition for two such pencils to be equivalent is that;