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Pb. 8.29 Find the multiplication rule for two 2 ⊗ 2 Hermitian matrices that
have been decomposed into the Pauli spin matrices and the identity matrix;
that is

If: M = a I + a σσ σσ + a σσ σσ + a σσ σσ
2 2
3 3
0
1 1
and N = b I + b σσ σσ + b σσ σσ + b σσ σσ
1 1
2 2
3 3
0
Find: the p-components in: P = MN = p I + p σσ σσ + p σσ σσ + p σσ σσ
0 1 1 2 2 3 3

Homework Problem

Pb. 8.30 The Calogero and Perelomov matrices of dimensions n ⊗ n are
given by:

 ( lk− )π 
−
M = (1 δ + j cot 
lk lk  ) 1  
  n 
a. Verify that their eigenvalues are given by:

λ = 2s – n – 1
s
where s = 1, 2, 3, …, n.
b. Verify that their eigenvectors matrices are given by:

 2π 
V = exp  j − ls
ls  n 

c. Use the above results to derive the Diophantine summation rule:

n−1
∑ cot   lπ  sin  2 slπ  = n − 2 s

l=1 n  n 
where s = 1, 2, 3, …, n – 1.

8.10.2 Unitary Matrices
DEFINITION A unitary matrix has the property that its Hermitian adjoint is
equal to its inverse:

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