# ch8 - CHAPTER 8 MATRIX EXPONENTIAL METHODS SECTION 8.1 In...

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416 Chapter 8 CHAPTER 8 MATRIX EXPONENTIAL METHODS SECTION 8.1 In Problems 1–8 we first use the eigenvalues and eigenvectors of the coefficient matrix A to find first a fundamental matrix Φ ( t ) for the homogeneous system x = Ax . Then we apply the formula x ( t ) = Φ ( t ) Φ (0) - 1 x 0 , to find the solution vector x ( t ) that satisfies the initial condition x (0) = x 0 . Formulas (11) and (12) in the text provide inverses of 2-by-2 and 3-by-3 matrices. 1. Eigensystem: TT 11 2 2 1, [1 1] ; 3, 1] λλ == = = vv 12 3 3 () tt ee te e  Φ= =  33 3 5 2 22 5 t t e e t + =⋅ =  −− + x 2. Eigensystem: 2 2 0, 2] ; 4, 2] 4 4 1 t t e e e = 44 21 2 13 5 21 1 6 1 0 t + = 3. Eigensystem: T 4, [1 2 2 ] ii λ + v cos4 2sin4 2cos4 sin4 ( ) R e I m t t e −+ = 0 2 0 5sin4 2 1 1 4cos4 t t t t t t = x

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Section 8.1 417 4. Eigensystem: TT 12 1 2 2, 2; { , } with [1 1] , 0] λ == = vv v v 2 11 2 () ( ) 1 tt t t te t e e t λλ +  Φ= + =  v 22 0 1 1 1 1 0 e ++   =⋅ =     x 5. Eigensystem: T 3, [ 1 3 ] ii + v cos3 sin3 ( ) R e I m 3cos3 3sin3 e −− = 0 1 1 3 1 1 6sin3 33 t t t =  + x 6. Eigensystem: T 54 , [ 12 2 ] =+ = + 5 cos4 2sin4 2cos4 ( ) R e I m t e e −+ = 5 5 0 2 2 sin4 1 2 2 4 0 4 t t t t e t + = x 7. Eigensystem: T 3 3 0 , [6 2 5 ]; 1 , [3 1 2 1 , [2 1 2 ] = = v 3 123 63 2 2 52 2 t ee e e e e = vvv 0 2 1 2 1 21 2 2 2 1 2 2 1 4 4 1 3 0 0 1 08 2 t t e e e + + = + + + + x 8. Eigensystem: T 2 2 3 3 2 , [0 1 1 1 , 1 0 3 , 1 1 ] =− = = = = =
418 Chapter 8 3 12 3 23 123 0 () 0 tt t ttt ee te e e e e e λ λλ  Φ= = −−  vvv 3 2 2 01 1 0 1 0 1 1 0 1 1 1 t t t t e e e e e e  =− = + x In each of Problems 9 - 20 we first solve the given linear system to find two linearly independent solutions x 1 and x 2 , then set up the fundamental matrix [] t = xx Φ , and finally calculate the matrix exponential 1 () (0 ) t e t = ΦΦ A . 9. Eigensystem: TT 11 2 2 1, [1 1] ; 3, [2 1] == = = vv 3 3 2 e = 33 3 3 22 2 2 2 t t t t t e e e e e −+ A 10. Eigensystem: 2 2 0, 2, [3 2] 2 2 13 t t e e e = 2 2 32 t t t e e e A 11. Eigensystem: 2, 1] ; [3 2] 3 2 e = 2323 3 3 3 2 3 2 tttt t eeee e A

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Section 8.1 419 12. Eigensystem: TT 11 2 2 1, [1 1] ; 2, [4 3] λλ == = = vv 12 2 2 4 () 3 tt ee te e  Φ= =
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## This note was uploaded on 04/08/2008 for the course MATH 374 taught by Professor Zhu during the Spring '08 term at Western Michigan.

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ch8 - CHAPTER 8 MATRIX EXPONENTIAL METHODS SECTION 8.1 In...

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