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MT_8-17 - Tutorial MT Matrix Theory = I R R2 R3 Rn S = S RS...

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Tutorial MT: Matrix Theory = ¡ I + R + R 2 + R 3 + ¢¢¢ + R n + ¢¢¢ ¢ S = S + RS + R 2 S + ¢¢¢ : The value of this solution will depend upon its convergence, in a sense that relates to the matrices involved. Exercise 8. One can see by inspection that the Pauli matrices are hermitian and that their traces vanish. They all have determinant ¡ 1 : That each is unitary can be seen by forming the products ¾ y i ¾ i : ¾ y 1 ¾ 1 = ¾ 1 ¾ y 1 = ¾ 2 1 = μ 0 1 1 0 ¶μ 0 1 1 0 = μ 1 0 0 1 ; ¾ y 2 ¾ 2 = ¾ 2 ¾ y 2 = ¾ 2 2 = μ 0 ¡ i i 0 ¶μ 0 ¡ i i 0 = μ 1 0 0 1 ; ¾ y 3 ¾ 3 = ¾ 3 ¾ y 3 = ¾ 2 3 = μ 1 0 0 ¡ 1 ¶μ 1 0 0 ¡ 1 = μ 1 0 0 1 : Furthermore, ¾ 1 ¾ 2 = μ 0 1 1 0 ¶μ 0 ¡ i i 0 = μ i 0 0 ¡ i = i μ 1 0 0 ¡ 1 = 3 ; ¾ 2 ¾ 1 = μ 0 ¡ i i 0 ¶μ 0 1 1 0 = μ ¡ i 0 0 i = ¡ i μ 1 0 0 ¡ 1 = ¡ 3 ; so that ¾ 1 ¾ 2 ¡ ¾ 2 ¾ 1 = [ ¾ 1 ; ¾ 2 ] = 2 3 . Similarly, ¾ 2 ¾ 3 = ¡ ¾ 3 ¾ 2 = 1 , ¾ 3 ¾ 1 = ¡ ¾ 1 ¾ 3 = 2 , [ ¾ 2 ; ¾ 3 ] = 2 1 and [ ¾ 3 ; ¾ 1 ] = 2 2 . Finally, we can consider the anticommutators : f ¾ i ; ¾ j g ´ ¾ i ¾ j + ¾ j ¾ i . From Eqns. 28 and 29, we have
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