6.3 - y = e λt we get from y 00 = 5 y 4 y that 2 = 5 4...

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Math 2940 Solutions, Fall 2011 Section 6 . 3 3 ) (a) Since we are talking about eigenvalues, we area assuming A is a square ( n × n ) matrix. If each column adds to 0, then the sum of the rows is the zero vector so the rows are dependent. Thus the rank of our matrix is at most n - 1 and it has 0 as an eigenvalue. (b) The matrix ± - 2 3 2 - 3 ² has characteristic polynomial λ 2 + 5 λ which has roots 0 , - 5 and eigenvectors (3 , 2) and (1 , - 1). We expand (4 , 1) as a linear combination of these to get (4 , 1) = (3 , 2) + (1 , - 1) and study e 0 t (3 , 2) and e - 5 t (1 , - 1). The solution (3 , 2) + e - 5 t (1 , - 1) has steady state (3 , 2). 6 ) The matrix ± a 1 1 a ² has characteristic polynomial λ 2 - 2 + ( a 2 - 1) which has roots a - 1 , a +1. We need these both to be negative for the solutions to go to zero. Thus we need a < - 1. The matrix ± b 1 - 1 a ² has characteristic polynomial λ 2 - 2 + ( b 2 + 1) which has roots b ± i . We need the real part of both to be negative for the solutions to go to zero. Thus we need b < 0. 10 ) A = ± 0 1 4 5 ² has characteristic polynomial λ 2 - 5 λ - 4 which has roots 5 ± 41 2 . On substituting
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Unformatted text preview: y = e λt , we get from y 00 = 5 y + 4 y that λ 2 = 5 λ + 4 giving the same roots. 17 ) (a) We need a matrix with both eigenvalues real, one positive an one negative. ±-1 0 0 1 ² works. (b) We need a matrix with both eigenvalues real and both positive. ± 1 0 0 1 ² works. (c) We need a matrix with both eigenvalues complex and real part positive. ± a b-b a ² works, where a > 0. 19 ) e Bt = I + ∑ ∞ k =1 t k B k /k ! = I + tB + + + ... = I + tB = ± 1-4 t 1 ² . It’s derivative is ±-4 ² while Be Bt = ±-4 ²± 1-4 t 1 ² = ±-4 ² . 25 ) ± 1 3 0 0 ² 2 = ± 1 3 0 0 ² . Note A 3 = A 2 A = AA = A . Similarly, A 4 = A 3 A = AA = A . Continuing on, A s = A s-1 A = AA = A . (This is called proof by induction ). So e At = I + ∑ ∞ k =1 t k A k /k ! = I + ∑ ∞ k =1 t k A/k ! = I + ( e t-1) A = ± e t 3( e t-1) ² ....
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This note was uploaded on 01/02/2012 for the course MATH 2940 at Cornell.

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