Density Matrix of Angular Momentum

Direction-dependent matrices / (a,p,y) might be transformed supposing there exist matrices r(a,p,y) and T (a,P,y) such that 

Matrices r(a,P) and T(y) and their inverses are obtained in the ease of spin quantum number 1, but in the ease of higher multiplieities the eorresponding matrices are patterned after those of spin quantum number 1. Matriees S and 5 are triangular and independent of direetion however, for / 1 the individual spin matrices must be ealeulated  

the eigenfunction matrices obtained from angular momentum M(a,%y) can be written as 

From this result an intrinsic eigenfunction matrix is kno to be equal to an eigenfunction matrix of M,. Finally, angular momentum M(a,P,y), havmg direction cosines eosa, eosp, and eosy, can be written as 

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