ON THE GROUP OF REAL LINEAR TRANSFORMATIONS 



WHOSE INVARIANT IS A REAL 



QUADRATIC FORM. 



By Henry Tabkr. 



Presented May 13, 1896. 



In what follows G will denote the group of real linear homogeneous 

 transformations of determinant +1 whose invariaut is the real quadratic 

 form of non-zero determinant, 



where Q is a real symmetric matrix.* 



Since the quadratic form J is real, the roots of its characteristic 

 equation are all real. In the Proceedings of the London Mathematical 

 Society, Vol. XXVL, page 376, I have shown that for n = 4 the group 

 of all proper linear homogeneous transformations, real and imaginary, 

 whose invariant is the real quadratic form J, provided the roots of the 

 characteristic equation of JF are not all of the same sign, contains a real 

 transformation that cannot be generated by the repetition of any infini- 

 tesimal transformation of this group; and that therefore, a fortiori^ 

 cannot be generated by the repetition of an infinitesimal transforma- 

 tion of group G. It follows that for w > 4 not every transformation of 

 group G can be generated by the repetition of an infinitesimal transfor- 

 mation of group G if the roots of the characteristic equation of J are not 

 all of the same sign. 



On the other hand, if the roots of the characteristic equation of jp 

 are all positive or all negative, every transformation of group G can be 

 generated by the repetition of an infinitesimal transformation of this 

 group, t 



* I employ the notation of Cayley's " Memoir on the Automorpliic Linear 

 Transformation of a Bipartite Quadric Function," Philosophical Transactions, 1858, 

 with tliis exception : the transverse of a matrix or linear transformation <p will be 

 denoted by . 



t If the roots of the characteristic equation of Jf are all of the same sign, group 

 G is isoinorphic with the group of real proper orthogonal substitutions ; and every 



