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MT_18-25

# MT_18-25 - Tutorial MT Matrix Theory 2 4 3 = I 1 ii 2 4 3 =...

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Tutorial MT: Matrix Theory = I μ 1 ¡ ® 2 2! + ® 4 4! ¡¢¢¢ + i μ ® ¡ ® 3 3! + ¢¢¢ = I cos ® + i sin ®: Exercise 16. We have ^ r ¢ ¾ =(sin μ cos ' ) ¾ 1 +(sin μ sin ' ) ¾ 2 + (cos μ ) ¾ 3 ; so that ( ^ r ¢ ¾ ) 2 = h (sin μ cos ' ) 2 + (sin μ sin ' ) 2 +cos 2 μ i I + ¡ sin 2 μ cos ' sin ' ¢ [ ¾ 1 ¾ 2 + ¾ 2 ¾ 1 ] +(sin μ cos μ cos ' )[ ¾ 1 ¾ 3 + ¾ 3 ¾ 1 ] +(sin μ cos μ sin ' )[ ¾ 2 ¾ 3 + ¾ 3 ¾ 2 ] : Since the ¾ i anticommute, the last three terms vanish and ( ^ r ¢ ¾ ) 2 = £ sin 2 μ ¡ cos 2 ' +sin 2 ' ¢ + cos 2 μ ¤ I = I : Exercise 17. We repeat the steps in Exercise 15 with ®¾ i replaced by ® ^ r ¢ ¾ . Since ( ^ r ¢ ¾ ) 2 = I , the same development obtains. Exercise 18. The term ‘‘secular’’ comes from early work in the study of planetary motions, and means ‘‘lasting a very long (but not eternal) time.’’ That this apt description of planetary orbits would become a part of the phraseology relating to linear equations is not difficult to imagine.

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