TABER. — TRANSFORMATION OF A REAL BILINEAR FORM. 309 



teristic equation of the transformation.* Finally, I show that each 

 transformation of the second kind of either of the families T is the 

 (2 m + l) th power of a transformation of that family for any odd ex- 

 ponent 2 m -f- 1. 



The theorems given above depend upon considerations relating to the 

 exponential function 



e°=l + U +~2l+ gT + ' ' ' 



of the matrix or linear substitution U. This series is convergent for 

 any finite matrix. We have 



(e*)- 1 = e~ u , 



(?) = e*; 

 and, if U x and U 2 are commutative, 



in particular, for any integer m , 



(e°) m = e' U . 



For any linear substitution T of non-zero determinant a polynomial 



U= S" p c p T p can be found such that T= e u . Let £ u &, . . . £„, re- 



o 

 spectively of multiplicity fx. x , fi 2 , ■ • • /*„» be the distinct roots of the 



characteristic equation of T ; let, moreover, 



Gl (T) = { -&-«" ) ' 



(i, h = 1, 2, . . . v & ^ 



^ (T) =G, (1) (T 7 ) . . . G/ M ' (T) G^ (T) . . . 0," (T 7 ); 



(< = 1, 2, . . . v) 



and let 



9 



/(P) = 2", I" log 6 + 2 m, V - 1 + -|r (2P- CD - "J 



i (r-c) 8 , ( _ irf -i i (^-r Hpm 



where m^ w 2 , etc., are integers. Then 



r= e nr) . 



* The determinant of a transformation of the first kind is positive. 



