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MT_26-31

# MT_26-31 - Tutorial MT Matrix Theory =(1 1]2(1 2 = 0 =!1 =...

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Tutorial MT: Matrix Theory = [(1 ¡ ! ) +1] 2 [(1 ¡ ! ) ¡ 2] = 0 = ) ! 1 = ¡ 1 ; ! 2 = ! 3 = 2 : The eigenvector for ! = ¡ 1 is obtained from 0 @ 2 ¡ 1 ¡ 1 ¡ 1 2 ¡ 1 ¡ 1 ¡ 1 2 1 A 0 @ v 1 v 2 v 3 1 A = 0 = ) v 1 = v 2 = v 3 : Thus, ^ V (1) = 1 p 3 0 @ 1 1 1 1 A : For ! = 2 , we have 0 @ ¡ 1 ¡ 1 ¡ 1 ¡ 1 ¡ 1 ¡ 1 ¡ 1 ¡ 1 ¡ 1 1 A 0 @ v 1 v 2 v 3 1 A = 0 = ) v 3 = ¡ v 1 ¡ v 2 with v 1 and v 2 undetermined. This means that any values for v 1 and v 2 can be taken. We can, for example, set v 2 = 0 : Then, v 3 = ¡ v 1 , and the normalized eigenvector is ^ V (2) = 1 p 2 0 @ 1 0 ¡ 1 1 A : Or, we can choose v 1 = 0 = ) ^ V (3) = 1 p 2 0 @ 0 1 ¡ 1 1 A : Exercise 26. Following the suggested steps, we write Hu i = ¸ i u i ; Hu j = ¸ j u j : followed by u y j Hu i = ¸ i u y j u i ; u y i Hu j = ¸ j u y i u j : Taking the complex conjugate of the second of these equations, we have ³ u y i Hu j ´ ¤ = ³ ¸ j u y i u j ´ ¤ = ¸ ¤ j u y j u i = u y j H y u i = u y j Hu i ; 20

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Tutorial MT: Matrix Theory or the pair, u y j Hu i = ¸ i u y j u i ; u y j Hu i = ¸ ¤ j u y j u i : Subtracting the second of these from the first, we get, as desired, 0 = ¡ ¸ i ¡
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MT_26-31 - Tutorial MT Matrix Theory =(1 1]2(1 2 = 0 =!1 =...

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