TABER. — TRANSFORMATION OP 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))i + 1)'* power of a transformation of that family for any odd ex- 

 ponent 2 771 + 1. 



The theorems given above depend upon considerations relating to the 

 exponential function 



e-=l+?7+ — +— + ... 



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

 any finite matrix. "We have 



(O = e^; 

 and, if Ui and U^ are commutative, 



in particular, for any integer m , 



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

 U =■ 2"p Cp T^ can be found such that T ^^ e^. Let ^i, 4> • • • C , re- 

 spectively of multiplicity fi^, /xo, . . . /x , be the distinct roots of the 

 characteristic equation of T ; let, moreover, 



(t, ^ = 1, 2, . . . V ^ ^ 



F, (T) = G/" (T) . . . Or'^ (T) Gr'^ {T) . . . Gl"' (T) ; 



(^■ = 1, 2, . . . v) 

 and let 



f(T) = r, ^ log C. + 2;.., V~=^l + -J- {T-Q - -^ ilz^ 



where mi, ?7?2, etc., are integers. Then 



T= e^^'K 

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



