MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 11 



We note that this is formed by writing zeros on the major diagonal, then 

 going back from the lower right corner writing ui, uo and us around the edges. 

 We then make the lower left term negative, then operate on the opposite 

 side of the major diagonal so as to make the matrix skew symmetric. 



The reciprocal of any matrix m is 



"'" \m\ ^*-^' 



where Mji is the // minor of \m\. 



The cross matrix has no reciprocal as for it (4.9) becomes indeterminate. 

 Since the vector product of two vectors ti and v is another vector per- 

 pendicular to both w and v and of a length uv sin (uv) we may write, on the 

 primed system 



Xi X .To 



''a2i\ / — 013^22 + ai2a23\ /az^ 



fl22 I = I 013021 — Ou023 I = I O32 



\an/ \ — O12O21 + OUO22/ \fl33y 



Matrices including vectors are equal only when their corresponding terms 

 are equal. Hence, we get the relations 



031 = O12O23 O13O22 



032 = O13O21 — O11O23 (4.10) 



033 ^ 011O22 — 012O21 



Similarly we get the relations: 



flu = 022033 — 023032 



O12 = O23O31 — 02lfl33 



fll3 = O21O32 — O22O3I (A^W 



021 = O32O13 — O12O33 



022 = O33OU — O31O13 



023 = fl3lOl2 ~ fl320u 



The 21 relations between the Ciy's allow us to complete the matrix given 

 four terms. 



Several Useful Matrix Relations 



/d/d x\ 



The del operator is the pseudo vector V = I d/d a-2 ) (4.12) 



\d/d Xz/ 



