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VI. Egenvektorer og diagonalisering – 2. Diagonalisering 460 Skrives også eller AB = BΛ B −1 AB = Λ 2.10. Diagonalisering og egenvektorer ☞ [LA] 10 Diagonalisering Sætning 16 Lad A vœre en n × n-matrix og b1,...,bn egentlige egenvektorer med tilhørende egenvœrdier λ1,...,λn. For matricen B, hvis søjler er egenvektorerne gœlder AB = BΛ hvor Λ er diagonalmatricen med egenvœrdierne som diagonalindgange. Hvis B er invertibel, vil den diagonalisere A. 2.11. Gammelt eksempel ☞ [LA] 9 Egenværdier og egenvektorer Eksempel 1, 2 Matricen har egenvektorer b1 = A = 3 3 −2 −4 −3 , b2 = 1 1 −2 1
VI. Egenvektorer og diagonalisering – 2. Diagonalisering 461 med tilhørende egenværdierne λ1 = 2,λ2 = −3. 2.12. Gammelt eksempel ☞ [LA] 9 Egenværdier og egenvektorer Eksempel 1, 2 - fortsat Dette giver A = som opfylder matrixidentiteten 3 3 , B = −2 −4 1 −3 −2 2 0 , Λ = 1 1 0 −3 AB = BΛ Da determinanten |B| = −5 2 er B invertibel og diagonaliserer A. 2.13. Matrixpotenser ☞ [LA] 10 Diagonalisering Eksempel - Potens Hvis B diagonaliserer A A = BΛB −1 så er potensen Λ k = A k = BΛ k B −1 ⎛ ⎜ ⎝ λ k 1 0 ... 0 . .. 0 . 0 λ k n ⎞ ⎟ ⎠
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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Forord "Slides" til forelæsningern
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