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VI. Egenvektorer og diagonalisering – 2. Diagonalisering 470 Opgave 2 - Løsning 2. Angiv en invertibel matrix B og en diagonal matrix Λ så at B −1 AB = Λ Søjler af egenvektorer giver ⎡ −1 B = ⎣ 1 −1 0 ⎤ 1 −1⎦, ⎡ 2 Λ = ⎣0 0 2 ⎤ 0 0 ⎦ 0 1 1 0 0 −1 det(B) = 1 sikrer invertibilitet. 2.31. Opgave, gør prøve Opgave 2 - Gør prøve! ☞ Matematik Alfa 1, August 2002 ⎡ −1 −3 A B = B Λ ⎤ ⎡ ⎤ −3 −1 −1 1 ⎡ −2 −2 ⎤ −1 ⎣ 3 5 3 ⎦ ⎣ 1 0 −1⎦ = ⎣ 2 0 1 ⎦ −3 −3 −1 0 ⎡ ⎤ ⎡ −1 −1 1 2 1 0 1 ⎤ 0 0 2 −1 ⎡ ⎤ −2 −2 −1 ⎣ 1 0 −1⎦ ⎣0 2 0 ⎦ = ⎣ 2 0 1 ⎦ 0 1 1 0 0 −1 0 2 −1 Så prøven stemmer! 2.32. Prikprodukt ☞ [S] 9.3 The dot product
VI. Egenvektorer og diagonalisering – 2. Diagonalisering 471 Definition b a Længden af en vektor a betegnes a. Prikproduktet, skalarproduktet af vektorer a,b er a · b = a bcos(θ) Udtrykt ved den vinkelrette projektion ba af b på a a · b = a ba 2.33. Prikprodukt ☞ [S] 9.3 The dot product I koordinater fås (b1, b2) a · b = a1b1 + a2b2 a = √ a · a = (a1, a2) a 2 1 + a2 2
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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