A PRINCIPAL AXIS TRANSFORMATION FOR NON-HERMITIAN MATRICES CARL ECKART AND GALE YOUNG The availability of the principal axis transformation for hermitian matrices often simplifies the proof of theorems

each two nonparallel elements of G cross each other. Obviously the conclusions of the theorem do not hold. The following example will show that the condition that no two elements of the collection G shall have a complementary domain in common is also necessary. In the cartesian plane let M be a circle of radius 1 and center at the origin, and iVa circle of radius 1 and center at the point (5, 5). Let d be a collection which contains each continuum which is the sum of M and a horizontal straight line interval of length 10 whose left-hand end point is on the circle M and which contains no point within M. Let G2 be a collection which contains each continuum which is the sum of N and a vertical straight line interval of length 10 whose upper end point is on the circle N and which contains no point within N. Let G = Gi+G2 . No element of G crosses any other element of G, but uncountably many have a complementary domain in common with some other element of the collection. However, it is evident that no countable subcollection of G covers the set of points each of which is common to two continua of the collection G. I t is not known whether or not the condition that each element of G shall separate some complementary domain of every other one can be omitted.