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Math 23

# Math 23 - 22 Using a computer matrix 2 4 4 A 1 2 3 —3...

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Unformatted text preview: 22. Using a computer, matrix 2 4 4 A: 1 2 3 —3 —4 —-5 has the following eigenvalue-eigenvector pairs. 0 «2 ——l—~—> ~1,2i——> —1——i, l 2 —2 ~2i ~—> *1 +i 2 Using Euler’s formula, . ~2 Z(t)=€2” __1__i 2 =(cosZt+isin2t) 1 ~2 0 x -—1 +i -—1 2 0 ‘2 0 = cosZt —l —sin2t —1 2 0 0 —2 +i cosZt -1 +sin2t *1 ‘ 0 2 The real and imaginary parts of z are solutions and we can write the general solution. 0 \ ’2 cosZt 3’0) = C164 “11) + C2 ~cosit + sian 1 2cos2t . 96 3. It is easily checked that A2: Therefore, the series GOO GOO GOO eA:I+A+_1-A2+... truncates and 10 0 eA=I+A2 0 1 (Hi 2 1 0 001 ~10 0 0. 01 A3 = AA2 —2 1 ~3 = -l 1 -1 1 «1 1 0 0 0 = 0 0 0 0 0 0 Thereivre in? series 2! 1 eA=I+A+—A2+... truncates and 2! eA=I+A+éA2 10 0 ’2 = 010 + -1 0 01 1 10 2 2 +— 0 1 1 2 0 -1 -1 —1 2 «2 = —1 5/2 “1/2. 1 —3/2 3/2 -—1 -—1 0 0 0 O 5 ‘i V) e ...
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