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final_review (dragged) 6 - MA 36600 FINAL REVIEW 7 §7.8:...

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Unformatted text preview: MA 36600 FINAL REVIEW 7 §7.8: Repeated Eigenvalues. • Consider 2 × 2 matrix A with eigenvalues r1 and r2 having corresponding eigenvectors ￿￿ ￿￿ ξ11 ξ (1) (2) ξ= and ξ = 12 . ξ21 ξ22 i. Say that r1 ￿= r2 i.e., we have distinct eigenvalues. Then there exist 2 × 2 matrices ￿ ￿ ￿ ￿ ξ ξ r 0 T = 11 12 and D= 1 such that T−1 A T = D. ξ21 ξ22 0 r2 ii. Say r1 = r2 i.e., we have repeated eigenvalues. If A = r1 I then set ￿ ￿ ￿ ￿ 10 r1 0 −1 T= =⇒ D = T AT = . 01 0 r1 If A ￿= r1 I, then choose a nonzero vector ￿￿ η η= 1 via the relation η2 (A − r1 I) η = ξ (1) . (This is a generalized eigenvector.) Now set ￿ ￿ ￿ ξ η1 r T = 11 =⇒ J = T−1 A T = 1 ξ21 η2 0 In particular, A is diagonalizable if and only if either (1) is a scalar multiple of the identity matrix. The matrices form for A. • For any r1 , r2 , and c, ￿ ￿ ￿ ￿ ￿rt r1 0 r1 c e1 D= , J= =⇒ exp (D t) = 0 r2 0 r1 0 ￿ 1 . r1 A has distinct eigenvalues or (2) A = r I D and J are called the Jordan canonical 0 er2 t ￿ , ￿ 1 exp (J t) = 0 ￿ c t r1 t e. 1 For any given 2 × 2 matrix A, there exists a nonsingular matrix T such that either ￿ ￿ D T exp (D t) T−1 if diagonalizable; −1 T AT = =⇒ exp (A t) = J T exp (J t) T−1 otherwise. §7.9: Nonhomogeneous Linear Systems. • Consider the nonhomogeneous system d x = A x + g(t) dt where A is a constant n × n matrix. The general solution is in the form x(t) = x(c) (t) + x(p) (t) in terms of the homogeneous and particular solutions ￿t x(c) (t) = Φ(t) c and x(p) (t) = Φ(t) Φ(τ )−1 g(τ ) dτ ; where Φ(t) = exp (A t) is a fundamental matrix. • Assume that A has n distinct eigenvalues r1 , r2 , . . . , rn . to rk , and consider the matrices ξ11 ξ12 · · · ξ1n r1 0 · · · ξ21 ξ22 · · · ξ2n 0 r2 · · · T= . D=. . . , . .. .. . . . . . . . . . . . . ξn1 ξn2 ··· 0 ξnn 0 ··· Denote ξ (k) as an eigenvector corresponding 0 0 . . . in terms of ξ (k) rn Then T−1 A T = D i.e., A is a diagonalizable matrix. Define the vectors y(t) = T−1 x(t) h(t) = T −1 g(t) =⇒ d y = D y + h(t). dt ξ1k ξ2k = . . . . ξnk ...
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