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2 MA 36600 LECTURE NOTES: MONDAY, APRIL 27 Matrix Exponentiation Revisited. We saw in the previous lecture that D = r 1 0 0 r 2 = exp ( D t ) = e r 1 t 0 0 e r 2 t . We show that J = r c 0 r = exp ( J t ) = 1 c t 0 1 e rt . To see why, recall the definition exp ( J t ) = k J k t k /k !. Observe that for nonnegative integers k , J k = r k k c r k 1 0 r k . We can see this from the first few examples: k = 0: J 0 = I = 1 0 0 1 k = 1: J 1 = r c 0 r k 2: J k = J k 1 · J = r k 1 ( k 1) c r k 2 0 r k 1 r c 0 r = r k k c r k 1 0 r k Hence we have the matrix exp ( J t ) = k =0 J k t k k ! = k r k t k /k ! k k c r k 1 t k /k ! 0 k r k t k /k ! . From the Taylor Series expansion e x = k x k /k ! we find that k =0 r k t k k ! = e rt , k =0 k c r k 1 t k k ! = k =1 c r k 1 t k ( k 1)! = c t · k =1 r k 1 t k 1 ( k 1)! = c t e rt . This gives exp ( J t ) = e rt c t e rt 0 e rt = 1 c t 0 1 e rt . 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 T 1 A T = D or T 1 A T = J . We have seen how to exponentiate both
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