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VI. Egenvektorer og diagonalisering – 2. Diagonalisering 458 Bevis Den j-te søjle i In er ej, så den j-te søjle i AIn er den j-te søjle i A. Aej = a •j 2.6. Diagonalmatrix ☞ [LA] 10 Diagonalisering Bemærkning En diagonalmatrix er en kvadratisk matrix, hvor indgange udenfor diagonalen alle er 0. Skrives skematisk ⎛ ⎞ λ1 0 ... ⎜ Λ = ⎜ . ⎝ 0 .. ⎟ 0 ⎟ ⎠ . 0 λn 2.7. Diagonalmatrix ☞ [LA] 10 Diagonalisering Bemærkning - fortsat For en diagonalmatrix Λ er det karakteristiske polynomium |Λ − λI| = (λ1 − λ) · · · (λn − λ) Egenvædierne er netop diagonal indgangene λ1,...,λn.
VI. Egenvektorer og diagonalisering – 2. Diagonalisering 459 Udregningen Λei = λiei viser at for egenværdien λi er ei en egentlig egenvektorer. 2.8. Simpel udregning ☞ [LA] 10 Diagonalisering Eksempel ⎛ 1 0 ⎞ ⎛ ⎞ 0 1 ⎛ ⎞ 1 ⎝0 2 0⎠ ⎝0⎠ = ⎝0⎠ ⎛ 1 ⎝0 0 0 3 0 0 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 2 0⎠ ⎝1⎠ = ⎝2⎠ = 2⎝1⎠ 0 ⎛ 1 0 0 3 0 ⎞ ⎛ ⎞ 0 0 0 ⎛ ⎞ 0 0 ⎛ ⎞ 0 ⎝0 2 0⎠ ⎝0⎠ = ⎝0⎠ = 3⎝0⎠ 0 0 3 1 3 1 2.9. At diagonalisere ☞ [LA] 10 Diagonalisering Definition At diagonalisere en kvadratisk matrix A vil sige at finde en invertibel matrix B og en diagonalmatrix Λ så A = BΛB −1
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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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1. Grafiske/numeriske metoder IV Di
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