TABER. — LINEAR TRANSFORMATION. 185 



For assuming that the determinant of the form is not zero, that is, 

 I O I 4^ 0, it follows that | </> | 4^ 0. Consequently a polynomial 

 X =^ f i4>) c^" be found such that 



Let 



then 



J. „'*" 

 <p ^ e 



From the identity 



e He ^= ilj 

 we have also 



Since 

 therefore 



is a polynomial in <f), and consequently commutative with &i2. 

 Whence we have * 



sa + Qx'i r?n n» / / — i ^ 

 e =e e = (fxp '^ = o. 



Conversely, if ^ O and fi ^9^ are commutative and such that 



then if 



</) = e , 



from the preceding identity it follows that 



* See note, page 183. 



t For any positive integer m we have 



Therefore 



t Since 



nf{4>) n-i = a (Sm cm <^'") n-i = s^c^ n<^'«n-i = Sm c^ {a <f> n-'^)'" 



= 2„c„(.^-i)'»=/(<|.-i). 



