8 BELL SYSTEM TECHNICAL JOURNAL 



The significance of the oT is that the matrix product of a and oT^ is a matrix 

 with ones on the major diagonal and all other terms zero. Whenever the 

 product of two square matrices gives such a matrix (known as the idem- 

 factor, /) they are said to be reciprocal. Only square matrices have 

 reciprocals. Multiplying any matrix by the idemfactor leaves the matrix 

 unchanged. We might consider, as part of our mathematical short hand, 

 that eq. (4.1) was solved for x by multiplying through by oT , as 



a~ y = a' ax = Ix = x. 



We must remember that the order must not be disturbed as the quantities 

 are not commutable, and that only square matrices have reciprocals. 



The major diagonal of a square matrix is the set of terms running diagon- 

 ally from the upper left to the lower right. 



A symmetrical matrix has any term Ma = Ma 



An anti-symmetric or skew symmetric matrix has any term Mij = — Mji 

 for i 9^ j. 



Rotation Theory 



The matrix algebra can be used to express a vector as a function of another 

 vector, that is to handle such relations as exist between E and P of section 2. 



There is another important aspect of matrix multiplication, that of trans- 

 forming a function from one set of axes to another. Let us assume that the 

 new set of unit axes, .ti .T2 and Xz are merely the old ones rotated through 

 angle <j) about some axis A which is a unit vector passing through the origin. 

 From Fig. Zi we see that in the expression: 



/ / / 



.Ti = Gii.Ti + a2iX2 -\- anXz 



the flij's are the cosines of the angles between .Ti and the three quantities 

 Xj. Conversely they are the cosines of the angles between the x/s, and :Vi. 

 Consequently, if the primed unit vectors are given in terms of the unprimed 

 ones by the three equations 



/ 



then the unprimed .v's are given in terms of the primed ones by the ex- 

 pression: 



/ 



This reversible relationship is well depicted by the table: 



