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lecture_36 (dragged) - MA 36600 LECTURE NOTES: WEDNESDAY,...

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Unformatted text preview: MA 36600 LECTURE NOTES: WEDNESDAY, APRIL 22 Fundamental Matrices Diagonalizable Matrices. We explain how to exponentiate an arbitrary matrix. First, we consider a special case. We say that an n × n matrix is a diagonal matrix if it is in the form r1 0 · · · 0 0 r2 · · · 0 D=. . .. . . . . . .. . . 0 0 · · · rn We will show that the exponential of a diagonal matrix is the diagonal matrix rt e1 0 ··· 0 0 er2 t · · · 0 exp (D t) = . . . . .. . . . . . . . 0 0 ern t ··· First, note that the product of two n × n diagonal matrices is again an n × n (1) (2) r1 0 ··· 0 r1 0 (1) (2) 0 0 r2 ··· 0 r2 D(1) = . and D(2) = . . . . .. . . . . . . . . . . . (1) 0 0 · · · rn 0 0 give the product D (1) D (2) (1) (2) r1 r1 0 = . . . 0 (1) (2) r2 r2 . . . ··· ··· .. . 0 ··· 0 In particular, we have r1 0 0 r2 D=. . . . . . 0 0 ··· ··· .. . ··· 0 0 . . . ∞ ￿ k=0 Dk tk = k! =⇒ 0 rn This gives the following matrix ￿ exp (D t) = k r1 0 Dk = . . . kk k r1 t /k ! 0 . . . 0 0 ··· 0 ··· kk k r2 t /k ! · · · . .. . . . ￿ (1) (2) rn rn ··· ··· .. . 0 ··· because we may set x = r t in the Taylor Series expansion ex = ∞ ￿ xk k=0 1 k! . ￿ 0 0 . . . ··· (2) rn 0 0 . . . 0 k r2 . . . diagonal matrix: ··· 0 ··· 0 . .. . . . . 0 0 . . . for k = 0, 1, 2, . . . . k rn rt e1 0 = . . . kk k rn t /k ! 0 0 er2 t . . . 0 ··· ··· .. . 0 0 . . . ··· ern t ...
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