A Mathematical Tools

Appendix A
Mathematical Tools

A.1 The Kronecker Product

For matrices $Q = (Qi,j)n,m ∈ ℝn ×m i,j=1$ and $R ∈ ℝp ×q$ , the Kronecker product is defined as the block matrix

Q⊗R =

( )
Q1,1R Q1,2R ... Q1,m −1R Q1,mR
|| Q2,1R Q2,2R ... Q2,m −1R Q2,mR ||
|| .. .. .. .. .. ||
| . . . . . |
( Qn −1,1R Qn −1,2R ... Qn −1,m−1R Qn −1,mR )
Qn,1R Qn,2R ... Qn,m −1R Qn,mR

∈ ℝ^np×mq .

The Kronecker product is bilinear and associative, but not commutative. Moreover, if the matrices $Q$ , $R$ , $S$ and $T$ are such that the products $QS$ and $RT$ can be formed, there holds

(Q⊗R)(S ⊗T) = (QS) ⊗ (RT) .

A direct consequence is that $Q ⊗ R$ is invertible if and only if $Q$ and $R$ are invertible. In this case, the inverse is given by

(Q⊗R)⁻¹ = Q⁻¹ ⊗R⁻¹ .

Suppose now that $n = m$ and $p = q$ , i.e. $Q$ and $R$ are square matrices. Let $λ1,...,λn$ denote the eigenvalues of $Q$ , and $μ1,...,μp$ denote the eigenvalues of $R$ . Then the eigenvalues of $Q ⊗ R$ are given by

λ_iμ_j, i = 1,…,n,j = 1,…,p .

A similar statement holds true for the singular values of general rectangular matrices $Q$ and $R$ . In particular, there holds

rank

Q⊗R

= rankQ× rankR .

A.2 Wigner 3jm Symbols

The symbol

( ) j1 j2 j3 m1 m2 m3 (A.1)

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:

$m1 ∈ {− |j1|,...|j1|}$ , $m2 ∈ {− |j2|,...|j2|}$ and $m3 ∈ {− |j3|,...|j3|}$ ,
$m1 + m2 + m3 = 0$ ,
$|j1 − j2| ≤ j3 ≤ j1 + j2$ .

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.

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Appendix AMathematical Tools

A.1 The Kronecker Product

A.2 Wigner 3jm Symbols

Appendix A
Mathematical Tools