sln3 - 7224 Homework Solution 3 Problem 1(a For a 2 2 real...

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7224 Homework Solution 3 Problem 1 (a) For a 2 × 2 real matrix A = ˆ a 11 a 12 a 21 a 22 ! The eigenvalues of the matrix A are given by λ = τ ± τ 2 - 4 δ 2 where τ = trace ( A ) and δ = det ( A ) = a 11 a 22 - a 12 a 21 . If δ > 0 , τ < 0 and τ 2 - 4 δ 0 then there are two real eigenvalues, which are negative. If δ > 0 , τ < 0 and τ 2 - 4 δ < 0 then there are two complex eigenvalues with negative real parts. Each discussion above is invertible. So they are necessary and sufficient conditions for A to have eigenvalues with negative real parts. (b) For a linear dynamical system d u dt = A u When det ( A ) > 0 and trace ( A ) < 0, there exist an invertible 2 × 2matrix P such that the matrix B = P - 1 AP has one of the following forms B = ˆ λ 1 0 0 λ 2 ! , B = ˆ λ 1 0 λ ! or B = ˆ a - b b a ! Where λ = a + ib is the complex eigenvalue in the third form. The system becomes d v dt = B v where v = P u is the linear transform of u . The eigenvalues ( λ 0 s ) of v are also eigenvalues of u . 1
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The solution of system could be written as u ( t ) = e Bt u 0 where e Bt has one of the following forms e Bt = ˆ e λ 1 0 0 e λ 2 ! , e Bt = e λ ˆ 1 t 0 1 !
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