l-7^r + f,x^-etc. 



TABER. — REAL LINEAR TRANSFORMATIONS. 79 



Let now e^ denote the infinite series 



1 +x + ^x'+3iX' + etc., 

 convergent for any Hnite matrix. We have 



(>) = e\ 

 and for any integer lu, 



moreover, if ^ and -^ are commutative, 



gX + X ^ e^ e^ = e^ e\ 



Corresponding to any finite matrix, ^, of non-zero determinant can be 



found a polynomial x i" ^ ^"^^ ^^^^ 



<j> = e^. 

 The infinite series 



4! 

 and 



are also convergent for any finite matrix, and are equal respectively to 



Therefore, if x and e^ ^~ are both real, the second power of the latter is 

 equal to the identical transformation. For if x and e are real, 



gxi^_g-x1'^^0; 

 that is, 



Since the determinant of (^i is not zero, by what precedes, a polynomial 

 X in <^i can be found such that 



and since — 1 is not a root of the characteristic equation of <^i, x niay be 

 so chosen that, if 



^ = x«"S 



we shall have 



^ = —&, 



