10 BELL SYSTEM TECHNICAL JOURNAL 



The squares of its terms must sum to zero, row by row as in (4.3) 



/ananan\ /aiia2i03i\ 

 I 021022023 1 ( O12O22O32 1 = 



\fl3lC32a33/ \Oi3023a33/ 



Because of the relations on + O12 + 013= 1, etc., we see that the terms of 

 the third matrix are zero for all terms not on the major diagonal. There- 

 fore, aac is an idemfactor and the reciprocal matrix of a is the same as its 

 conjugate matrix. 



ac = a~' (4.4) 



Also .vi is of unit length, and the sum of the squares of its components on 

 the unprimed system is unity. Thus we find: 



Oil + O21 + O31 = 1 



o?2 + 022 + O32 = 1 (4.5) 



Ol3 + O23 + O33 = 1 



We now introduce from vector analysis the concept of the scalar product. 

 The scalar product of two vectors u and v is Uc v. It is the product of the 

 lengths of the two vectors and the cosine of the angle between them. 



If we take the scalar product of .Ti and .T2 as expressed in the primed 

 system we have, since they are mutually perpendicular: 



/02l\ 



(an, O12, O13) I O22 1 = OnOoi + O12O22 + O13O23 = 



\023/ 



Similarly multiplying .V2 and .V3 scalarly, and .V3 and .Vi we find: 



O11O21 + 012O22 + O13O23 = 



O21O31 + O22O32 + 023033 = (4.5) 



O31O11 + O32O12 + O33O13 = 



If we multiply Xi and .V2 etc. as expressed on the unprimed system we get 

 the relations: 



011O12 + 021O22 + 031O32 = 



O12O13 + 022023 + O32O33 = (4.7) 



013011 + ^23021 + O33O31 = 



The vector product of two vectors u and v requires the defining of a 

 special matrix, the cross matrix. 



(0 — «3 «2\ 

 m -«ij (4.8) 



— U2 «1 0/ 



