lecture_38 (dragged) 1

lecture_38 (dragged) 1 - 2 MA 36600 LECTURE NOTES MONDAY...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 MA 36600 LECTURE NOTES: MONDAY, APRIL 27 Matrix Exponentiation Revisited. We saw in the previous lecture that ￿ ￿ ￿rt ￿ r1 0 e1 0 D= =⇒ exp (D t) = . 0 r2 0 er2 t We show that ￿ ￿ 1 c t rt =⇒ exp (J t) = e. 01 ￿ To see why, recall the definition exp (J t) = k Jk tk /k !. Observe that for nonnegative integers k , ￿k ￿ r k c rk−1 Jk = . 0 rk J= ￿ r 0 c r ￿ We can see this from the first few examples: ￿ ￿ 10 0 k = 0: J = I = 01 ￿ ￿ rc 1 k = 1: J = 0r ￿ k−1 r k k−1 k ≥ 2: J = J ·J= 0 (k − 1) c rk−2 rk−1 Hence we have the matrix ￿￿ r 0 ￿ ￿k c r = r 0 k c rk−1 rk ￿ ￿￿ k k ￿ ￿ k−1 k tk k r t /k ! k kcr ￿ k k t /k ! . exp (J t) = J = 0 k! k r t /k ! k=0 ￿k From the Taylor Series expansion ex = k x /k ! we find that ∞ ￿ ∞ ￿ k=0 rk tk = ert , k! This gives ∞ ￿ k k=0 ∞ ∞ k=1 k c rk−1 k=1 ￿ ￿ tk tk tk−1 = c rk−1 = ct · rk−1 = c t ert . k! (k − 1)! (k − 1)! ￿ rt ￿￿ ￿ e c t ert 1 c t rt = e. 0 ert 01 We explain how this is useful. Let A be a 2 × 2 matrix with constant entries. Denote D or J as the Jordan canonical form for A. Then there exists a nonsingular matrix T such that either exp (J t) = T−1 A T = D or T−1 A T = J. We have seen how to exponentiate both D and J. Hence we can exponentiate A very easily: ￿ T exp (D t) T−1 if disc A ￿= 0; exp (A t) = T exp (J t) T−1 if disc A = 0. Example. Consider again the 2 × 2 matrix A= We will compute exp (A t). We found before that if we denote ￿ ￿ 1 0 T= −1 −1 We also see that ￿ 2 J= 0 This gives the matrix exponential and ￿ 1 2 exp (A t) = T exp (J t) T−1 ￿ ￿￿ ￿￿ 1 01t 1 = −1 −1 0 1 −1 ￿ 1 1 ￿ 2 J= 0 =⇒ ￿ −1 . 3 1 2 ￿ =⇒ ￿ 1 exp (J t) = 0 ￿ ￿ 0 2t 1 e= −1 −1 ￿￿ 0 1−t −1 −1 T−1 A T = J. ￿ t 2t e. 1 ￿ ￿ −t 2t 1−t e= −1 t ￿ −t e2t . 1+t ...
View Full Document

This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

Ask a homework question - tutors are online