As is well known, a dyadic (Gibbs) has been regarded to be the same 

 as a matrix — or at least a special notation for a matrix. Bôcher says' : 

 "A matrix of rank one has been called bv Gimhs a dyad, since it may 

 be regarded as a product of two complex quantities {a^, a.,, . . . a,i) and 

 {bj^, /;.,, . . . /)„). The sum of any number of d3'ads is called a dyadic poly- 

 nomial, or simply a dyadic. Every matrix is therefore a dyadic, and vice 

 versa. Gibbs's theory of dyadics, in the case ;/ = 3, is explained in the 

 Vector Analysis of Gibbs-Wilsox, Chap. V. Geometric language is here used 

 exclusively, the complex quantities ia^, a.,, (7.^) and (/»j, a.,, /;.j) from which 

 the dyads are built up being interpreted as vectors in space of three 

 dimensions". 



It may be questioned whether this statement exactly holds for all 

 cases. Certainly Gibbs has not considered his vectors as being merely 

 geometric interpretations of complex quantities, but as primary objects. 

 His typical vector is the translation in space, thus a "thing" independent 

 of every particular coordinate system chosen. And the characteristic pro- 

 perty of his dyadic is that it converts one vector into another, thus not 

 a geometric interpretation of a matrix, but a notation for a transformaticMi 

 in space. The object of this note is to show that there may be cases 

 where the dvadic is a much more suitable notation for such a transfor- 

 mation than a matrix. 



Using 71 dimensions instead of the Gibbsian three-dimensional space, 

 and denoting the normal S3'stem of (orthogonal) unit \-ectors (which repre- 

 sents the coordinate system chosen) by e^, C.,, . . . e«, any dyadic can be 

 expressed in the following form 



(1) <P = e,a,- 



where a, is a set of vectors (summation with respect to a subscript ap- 

 pearing twice is always understood). Then if this dyadic transforms the 

 vector V into the vector v' , this is written 



(2) 0- 1> - e,a, • = v' 

 the dot denoting the scalar product. 



' Introduction to Higher Algebra p. 79. 



