Appendix A
Mathematical Tools
A.1 The Kronecker Product
For matrices and , the Kronecker product is defined as the
block matrix
Q⊗R = ∈ ℝ^{np×mq} .   
The Kronecker product is bilinear and associative, but not commutative. Moreover, if the
matrices , , and are such that the products and can be formed, there
holds
(Q⊗R)(S ⊗T) = (QS) ⊗ (RT) .   
A direct consequence is that is invertible if and only if and are invertible. In this
case, the inverse is given by
(Q⊗R)^{−1} = Q^{−1} ⊗R^{−1} .   
Suppose now that and , i.e. and are square matrices. Let
denote the eigenvalues of , and denote the eigenvalues of . Then the
eigenvalues of are given by
λ_{i}μ_{j}, i = 1,…,n,j = 1,…,p .   
A similar statement holds true for the singular values of general rectangular matrices and
. In particular, there holds
rankQ⊗R = rankQ× rankR .   
A.2 Wigner 3jm Symbols
The symbol
with
parameters being either integers or halfintegers is called a Wigner 3jm symbol arising in coupled
angular momenta between two quantum systems. It is zero unless all of the following selection
rules apply:
 , and ,
 ,
 .
The connection with spherical harmonics is the following:
∫
_{Ω}Y _{l1,m1}Y _{l2,m2}Y _{l3,m3}dΩ =   

 ×× ,   
where the left hand side is often termed Slater integral.