ch8 - CHAPTER 8 MATRIX EXPONENTIAL METHODS SECTION 8.1 In Problems 18 we first use the eigenvalues and eigenvectors of the coefficient matrix A to find

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: T T 1 1 2 2 1, [1 1] ; 3, [1 1] λ λ = = = = v v 1 2 3 1 2 3 ( ) t t t t t t e e t e e e e λ λ Φ = = v v 3 3 3 3 1 1 3 5 1 1 ( ) 1 1 2 2 2 5 t t t t t t t t e e e e t e e e e +   = =   +   x 2. Eigensystem: T T 1 1 2 2 0, [1 2] ; 4, [1 2] λ λ = = = = v v 1 2 4 1 2 4 1 ( ) 2 2 t t t t e t e e e λ λ Φ = = v v 4 4 4 4 2 1 2 1 3 5 1 1 ( ) 2 1 1 4 4 2 2 6 10 t t t t e e t e e +   = =     x 3. Eigensystem: T 4 , [1 2 2] i i λ = = + v cos4 2sin4 2cos4 sin4 ( ) Re( ) Im( ) 2cos4 2sin 4 t t t t t t t e e t t λ λ + Φ = = v v cos4 2sin4 2cos4 sin4 0 2 0 5sin 4 1 1 ( ) 2cos4 2sin4 2 1 1 4cos4 2sin4 4 4 t t t t t t t t t t +    = =       x
Section 8.1 417 4. Eigensystem: T T 1 2 1 2 2, 2; { , } with [1 1] , [1 0] λ = = = v v v v 2 1 1 2 1 1 ( ) ( ) 1 t t t t t e t e e t λ λ + Φ = + = v v v 2 2 1 1 0 1 1 1 ( ) 1 1 1 0 t t t t t e e t t + +      = =           x 5. Eigensystem: T 3 , [ 1 3] i i λ = = − + v cos3 sin3 cos3 sin3 ( ) Re( ) Im( ) 3cos3 3sin3 t t t t t t t e e t t λ λ Φ = = v v cos3 sin3 cos3 sin3 0 1 1 3cos3 sin3 1 1 ( ) 3cos3 3sin3 3 1 1 3cos3 6sin3 3 3 t t t t t t t t t t t   = =   +   x 6. Eigensystem: T 5 4 , [1 2 2] i i λ = + = + v 5 cos4 2sin4 2cos4 2sin4 ( ) Re( ) Im( ) 2cos4 2sin 4 t t t t t t t t e e e t t λ λ + Φ = = v v 5 5 cos4 2sin4 2cos4 2sin4 0 2 2 cos4 sin 4 1 ( ) 2 2cos4 2sin 4 2 4 0 sin4 4 t t t t t t t t t e e t t t + +    = =       x 7. Eigensystem: T T T 1 1 2 2 3 3 0, [6 2 5] ; 1, [3 1 2] ; 1, [2 1 2] λ λ λ = = = = = − = v v v 3 1 2 1 2 3 6 3 2 ( ) 2 5 2 2 t t t t t t t t t e e t e e e e e e e λ λ λ Φ = = v v v 6 3 2 0 2 1 2 12 12 2 ( ) 2 1 2 2 1 4 4 5 2 2 1 3 0 0 10 8 2 t t t t t t t t t t t t e e e e t e e e e e e e e + +       = = + +   

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