Complex, distinct eigenvalues (Sect. 7.6)

Transcription

1 Comple, distinct eigenvlues (Sect 76) Review: Clssifiction of 2 2 digonlizle systems Review: The cse of digonlizle mtrices Rel mtri with pir of comple eigenvlues Phse portrits for 2 2 systems Review: Clssifiction of 2 2 digonlizle systems Remrk: Digonlizle 2 2 mtrices A with rel coefficients re clssified ccording to their eigenvlues () λ λ 2, rel-vlued Hence, A hs two non-proportionl eigenvectors v, v 2 (eigen-directions), (Section 75) () λ = λ 2, comple-vlued Hence, A hs two non-proportionl eigenvectors v = v 2, (Section 76) (c-) λ = λ 2 rel-vlued with two non-proportionl eigenvectors v, v 2, (Section 77) Remrk: (c-2) λ = λ 2 rel-vlued with only one eigen-direction Hence, A is not digonlizle, (Section 77)

2 Comple, distinct eigenvlues (Sect 76) Review: Clssifiction of 2 2 digonlizle systems Review: The cse of digonlizle mtrices Rel mtri with pir of comple eigenvlues Phse portrits for 2 2 systems Review: The cse of digonlizle mtrices Theorem (Digonlizle mtri) If n n mtri A is digonlizle, with linerly independent eigenvectors set {v,, v n } nd corresponding eigenvlues {λ,, λ n }, then the generl solution to the homogeneous, constnt coefficients, liner system (t) = A (t) is given y the epression elow, where c,, c n R, (t) = c v e λ t + + c n v n e λ nt

3 Comple, distinct eigenvlues (Sect 76) Review: Clssifiction of 2 2 digonlizle systems Review: The cse of digonlizle mtrices Rel mtri with pir of comple eigenvlues Phse portrits for 2 2 systems Rel mtri with pir of comple eigenvlues Theorem If {λ, v} is n eigen-pir of n n n rel-vlued mtri A, then {λ, v} lso is n eigen-pir of mtri A Proof: By hypothesis A v = λ v nd A = A Then A v = λ v A v = λ v A v = λ v Therefore {λ, v} is n eigen-pir of mtri A Remrk: The Theorem ove is equivlent to the following: If n n n rel-vlued mtri A hs eigen pirs λ = α + iβ, with α, β R nd, R n, then so is λ 2 = α iβ, v = + i, v 2 = i

4 Rel mtri with pir of comple eigenvlues Theorem (Comple pirs) If n n n rel-vlued mtri A hs eigen pirs λ ± = α ± iβ, v (±) = ± i, with α, β R nd, R n, then the differentil eqution (t) = A (t) hs linerly independent set of two comple-vlued solutions (+) = v (+) e λ +t, ( ) = v ( ) e λ t, nd it lso hs linerly independent set of two rel-vlued solutions () = [ cos(βt) sin(βt) ] e αt, (2) = [ sin(βt) + cos(βt) ] e αt Rel mtri with pir of comple eigenvlues Proof: We know tht one solution to the differentil eqution is (+) = v (+) e λ +t = ( + i) e (α+iβ)t = ( + i) e αt e iβt Euler eqution implies (+) = ( + i) e αt [ cos(βt) + i sin(βt) ], (+) = [ cos(βt) sin(βt) ] e αt + i [ sin(βt) + cos(βt) ] e αt A similr clcultion done on ( ) implies ( ) = [ cos(βt) sin(βt) ] e αt i [ sin(βt) + cos(βt) ] e αt Introduce () = ( (+) + ( ) )/2, (2) = ( (+) ( ) )/(2i), then () = [ cos(βt) sin(βt) ] e αt, (2) = [ sin(βt) + cos(βt) ] e αt

5 Rel mtri with pir of comple eigenvlues Find rel-vlued set of fundmentl solutions to the eqution 2 3 = A, A = Solution: () Find the eigenvlues of mtri A ove, p(λ) = det(a λ I ) = (2 λ) 3 3 (2 λ) = (λ 2)2 + 9 The roots of the chrcteristic polynomil re (λ 2) = λ ± 2 = ±3i λ ± = 2 ± 3i (2) Find the eigenvectors of mtri A ove For λ +, 2 (2 + 3i) 3 A λ + I = A (2 + 3i)I = (2 + 3i) Rel mtri with pir of comple eigenvlues Find rel-vlued set of fundmentl solutions to the eqution 2 3 = A, A = 2 (2 + 3i) 3 Solution: λ ± = 2 ± 3i, (A λ + I ) = (2 + 3i) We need to solve (A λ + I ) v (+) = for v (+) Guss opertions 3i 3 i i i 3 3i i i So, the eigenvector v (+) v = is given y v v = iv 2 Choose 2 v 2 =, v = i, v (+) i =, λ + = 2 + 3i

6 Rel mtri with pir of comple eigenvlues Find rel-vlued set of fundmentl solutions to the eqution 2 3 = A, A = i Solution: Recll: eigenvlues λ ± = 2 ± 3i, nd v (+) = The second eigenvector is v ( ) = v (+), tht is, v ( ) = Notice tht v (±) = ± i The nottion λ ± = α ± βi nd v (±) = ± i implies α = 2, β = 3, =, = [ i ] Rel mtri with pir of comple eigenvlues Find rel-vlued set of fundmentl solutions to the eqution 2 3 = A, A = Solution: Recll: α = 2, β = 3, =, nd = Rel-vlued solutions re () = [ cos(βt) sin(βt) ] e αt, nd (2) = [ sin(βt) + cos(βt) ] e αt Tht is () = ( cos(3t) ) sin(3t) e 2t () = sin(3t) e 2t cos(3t) (2) = ( sin(3t)+ ) cos(3t) e 2t (2) = [ cos(3t) sin(3t) ] e 2t

7 Comple, distinct eigenvlues (Sect 76) Review: Clssifiction of 2 2 digonlizle systems Review: The cse of digonlizle mtrices Rel mtri with pir of comple eigenvlues Phse portrits for 2 2 systems Phse portrits for 2 2 systems Sketch phse portrit for solutions of = A, A = 2 3 Solution: The phse portrit of the vectors () sin(3t) =, cos(3t) (2) = cos(3t), sin(3t) () 2 (2) is rdius one circle

8 Phse portrits for 2 2 systems Sketch phse portrit for solutions of = A, A = 2 3 Solution: The phse portrit of the solutions () sin(3t) = e 2t, cos(3t) (2) = cos(3t) e 2t, sin(3t) re outgoing spirls () 2 (2) Phse portrits for 2 2 systems Given ny vectors nd, sketch qulittive phse portrits of () = [ cos(βt) sin(βt) ] e αt, (2) = [ sin(βt) + cos(βt) ] e αt for the cses α =, α >, nd α <, where β > Solution: 2 2 (2) 2 (2) () () () (2)