MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 21 



Using (1) this becomes 



P =l^(k-I)E (6.7) 



SECTION 7 



Quadratic Forms 



Often the elements of a matrix are themselves functions of other quanti- 

 ties. In order to relate the elements of one matrix with those of another by 

 means of a matrix multiplication, we may make a single column matrix of 

 each of them. We then wish to know how a transformation of axes changes 

 the elements of this single column matrix. Consider a symmetrical matrix 

 b that relates two vectors u and v: 



u = bv. 



A transformation of axes, a, changes u and v to u' and v'. Multiplying w = 

 bv through by the prefactor a we have 



au = abv. 



We now replace ate by its equivalent a~ v' whence: 



u' = abcT v' 



so that 



U — V 



if we define b' as 



b' = aba~' (7.1) 



To be in accord with common usage we now rearrange b according to the 

 arbitrary scheme: 



We wish to know what operation to perform on B to get B' corresponding 

 to b' . If we expand b' = aba" it is easily seen that b' = oB where 



