## Appendix AMathematical Tools

### A.1 The Kronecker Product

For matrices and , the Kronecker product is defined as the block matrix

 Q⊗R = ∈ ℝnp×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

 λiμj, 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 half-integers 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:
1. , and ,
2. ,
3. .

The connection with spherical harmonics is the following:

 ∫ ΩY l1,m1Y l2,m2Y l3,m3dΩ = ×× ,
where the left hand side is often termed Slater integral.