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VI. Egenvektorer og diagonalisering – 1. Egenvektorer 442 1.14. Karakteristisk polynomium ☞ [LA] 9 Egenværdier og egenvektorer Definition - skematisk Det karakteristiske polynomium af en n × n-matrix A er n-te grads polynomiet a11 − λ a12 · · · a1n a12 a22 − λ · · · a2n . . . . . .. . . an1 an2 · · · ann − λ = (−1) n λ n + · · · + |A| = |A − λIn| 1.15. Karakteristisk polynomium ☞ [LA] 9 Egenværdier og egenvektorer Eksempel Det karakteristiske polynomium af en 2 × 2-matrix A er andengrads polynomiet a11 − λ a12 a21 a22 − λ = (a11 − λ)(a22 − λ) − a12a21 = λ 2 − (a11 + a22)λ + (a11a22 − a12a21)
VI. Egenvektorer og diagonalisering – 1. Egenvektorer 443 1.16. Trekantsmatrix Eksempel ☞ [LA] 9 Egenværdier og egenvektorer Udregningen a11 − λ 0 . 0 a12 a22 − λ . 0 · · · · · · . .. · · · a1n a2n . . ann − λ = (a11 − λ)(a22 − λ) · · · (ann − λ) viser at egenværdierne i en trekantsmatrix netop udgøres af diagonal indgangene. 1.17. Egengenrum ☞ [LA] 9 Egenværdier og egenvektorer Sætning 15 Lad A vœre en n × n-matrix og λ en egenvœrdi. Så er mœngden af egenvektorer for A et lineœrt underrum af R n . Dette kaldes egenrummet hørende til λ og betegnes Eλ Bevis Egenrummet er løsningsrummet for det homogene ligningssystem med koefficientmatrix A − λIn. 1.18. Andengradsligning ☞ [LA] 9 Egenværdier og egenvektorer
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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