L98 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.8 The Matrix Exponential Function Wed, 03/Dec c ± 2003, Art Belmonte Summary In the following, let A and B be (real) n × n constant matrices, let r be an eigenvalue of A ,andlet v be an eigenvector associated with r (unless otherwise stated). Also, let s and t be scalars. Definitions The exponential of the matrix A is defined by the power series e A = I + A + 1 2! A 2 + 1 3! A 3 +···= X k = 0 1 k ! A k . Similarly, e t A = I + t A + t 2 2! A 2 + t 3 3! A 3 X k = 0 t k k ! A k . If ( A - r I ) p w = 0 , the zero vector, for some integer p 1, then w is called a generalized eigenvector of A . Properties of the matrix exponential e A 0 = e 0 = I , the identity matrix. e ( s + t ) A = e s A e t A . ( e t A ) - 1 = e - t A . If A and B commute (if AB = BA ), then e t ( A + B ) = e t A e t B . e st I = e I . Facts 1. d dt e t A = A e t A . (Differentiate term-by-term to see this.) 2. Let x 0 R n .Then x ( t ) = e t A x 0 is the unique solution to the initial value problem x 0 = Ax , x ( 0 ) = x 0 . 3. If r v is an eigenpair, then e t A v = e rt v . 4. A commutes with e A ; i.e., A e A = e A A . 5. If r is the only eigenvalue of A , then for some integer m < n we have e t A = e m X k = 0 t k k ! ( A - r I ) k . 6. If r is an eigenvalue of A with algebraic multiplicity q ,then there is a positive integer p q such that the dimension of the nullspace of ( A - r I ) p is equal to q . 7. Let X = X ( t ) and Y = Y ( t ) be fundamental matrices for u 0 = Au and define X 0 = X ( 0 ) . X = YC for some constant matrix C . e t A = X ( t ) X - 1 0 : This gives another way to compute e t A . The exponential matrix e t A itself is a fundamental matrix! Full load of generalized eigenvectors In light of #6, we finally have a procedure for always generating a fundamental solution set to x 0 = Ax . For each eigenvalue r of A , let q be the algebraic multiplicity of r . Do the following. Find the smallest positive integer p such that the nullspace of ( A - r I ) p has dimension q . Find a basis ± v 1 ,... v q ² for said nullspace. For each generalized eigenvector v j , 1 j q , construct the solution x j ( t ) = e t A v j = e p - 1 X k = 0 t k k ! ( A - r I ) k v j . If x j is associated with a complex eigenvalue r = α + i β having positive imaginary part (i.e., β> 0), take its real and imaginary parts to obtain two real solutions. (Toss out the conjugate eigenvalue α - i β .) The collection of all these solutions over the selected eigenvalues of A gives a real fundamental solution set. MOAF (The Mother of All Formulas) The solution of the IVP x 0 = Ax + f , x ( t 0 ) = x 0 , provided A is a constant matrix, is given by x ( t ) = e ( t - t 0 ) A x 0 + Z t t 0 e ( t - w) A f (w) d w. If t 0 = 0, this simplifies to x ( t ) = e t A x 0 + Z t 0 e ( t - A f (w) d 1
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“Hand” Examples Example A Given A = ± 11 - 1 - 1 ² , calculate e A via the definition. Solution Now A 2 = ± 00 ² , whence e A = I + A = ± 21 - 10 ² .
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L98 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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