1922. No. 13. ON A SPECIAL POLYADIC OF ORDER ;/ — p. 7 



Given a linear transformation, — the matrix of which is aij, i and 

 j = \, 2 . . . . )i — which transforms a vector t» = e,- tv into a vector 

 ü' = e,- I'i as follows ; 



t'l = «11 î'i + «12 r., + . . . . -f «1 H ï'h = a^iVi 



, . Vo = «Ol t'l + aoo î'o + . . . . + a.2 „ v„ = a^ i Vi 



Vu = a„i I'l -f- (hi.2 r.i + . . . . + Onii Vn = dniV, 



If we here introduce n vectors a^, Ct., . . . . Cl„ defined by 

 ak = Cj aki + C, Oko +....+ e„ak„ = ^iOki 

 we see that (3) can simply be written: 



I'/ = a, ■ r 



(4) 



v,i' = a« • i^ 



(5) or, briefly: I'k = Clj • V 

 and accordingly : 



(6) t>' = Ci a^ • y - e.., a., • r + . . . . + e« a„ • v = e, a, • i\ 

 The expression : 



(7) e, a, = Ci ill - e._> a. - .... ^ e„ a„ = .4 



is a dyadic in S«. It is completely defined by the vector system a,. The 

 vectors e, and Ck, in e, a, are said to be multiplied iiidctermmately with one 

 another. A dyad is the indeterminate product of any two vectors, its 

 corresponding matrix is of rank one, a dyadic is a sum of dyads. It is 

 frequently in literature called a tensor of the second order. 



Thus A • 'C is nothing but a linear transformation, and the matrix of 

 A or of the vector system a, is simpl}' the matrix of the transformation. 



We also call to memor}- that a dyadic can be resolved into a sum 

 of elementary dN'ads, i. e. indeterminate products of two unit vectors multi- 

 plied by a scalar factor. This is obtained by putting 



a, = ey Oij 



and expanding according to the distributive law of multiplication. 

 Therefore : 



(8) A = tiZjOij i,j=\.2 ... « 



As it is immaterial to which vector the scalar factor is applied, this 

 evidentlv mav be written: 



