# W17_M217_HW11_Sols - HOMEWORK 11 SOLUTIONS Part A(15 points...

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det 2 - λ - 2 0 0 1 - 1 - λ 0 0 0 0 3 - λ - 4 0 0 2 - 3 - λ = det 2 - λ - 2 1 - 1 - λ det 3 - λ - 4 2 - 3 - λ = ((2 - λ )( - 1 - λ ) + 2)((3 - λ )( - 3 - λ ) + 8) = ( - λ + λ 2 )( λ 2 - 1) = λ ( λ + 1)( λ - 1) 2 λ = 1 λ = - 1 λ = 0 det a - λ b b a - λ = ( a - λ 2 ) - b 2 = λ 2 - 2 + a 2 - b 2 λ = 2 a ± 4 a 2 - 4 a 2 + 4 b 2 2 = a ± b det 1 - λ 1 0 0 2 - λ 2 0 0 3 - λ = (1 - λ )(2 - λ )(3 - λ ) , 1 , 2 , 1 E 1 = ker 0 1 0 0 1 2 0 0 2 = span 1 0 0 ,
1 1 E 2 = ker - 1 1 0 0 0 2 0 0 1 = span 1 1 0 , 1 2 E 3 = ker - 2 1 0 0 - 1 2 0 0 0 = span 1 2 1 , 1 3 det 1 - λ 0 0 - 5 - λ 2 0 0 1 - λ = - λ (1 - λ ) 2 , 1 1 E 0 = ker 1 0 0 - 5 0 2 0 0 1 = span 0 1 0 , 1 0 E 1 = ker 0 0 0 - 5 - 1 2 0 0 0 = span 0 2 1 , 2 / 5 0 1 , 2 1 det 1 - λ a 0 b - λ = ( λ - 1)( λ - b ) , 1 b 1 b 6 = 1 b = 1 ker 0 a 0 0 = span (" 1 0 #) ; if a 6 = 0 span (" 1 0 # , " 0 1 #) ; if a = 0 b = 1 a = 0 5 f 0 - 3 f = λf f 0 = 1 5 (3+ λ ) f λ f ( x ) = e 1 5 (3+ λ ) x λ
xf 0 ( x ) = λf ( x ) na n x n + ( n - 1) a n - 1 x n - 1 + · · · + 2 a 2 x 2 + a 1 x = λa n x n + λa n - 1 x n - 1 + · · · + λa 2 x 2 + λa 1 x + λa 0 λ λ = k x k det λ - 1 0 0 λ - 1 - 5 7 λ - 3 = λ det λ - 1 7 λ - 3 + det 0 - 1 - 5 λ - 3 = λ ( λ 2 - 3 λ + 7) - 5 = λ 3 - 3 λ 2 + 7 λ - 5 = ( λ - 1)( λ 2 - 2 λ + 5) , 1 1 + 2 i 1 - 2 i 1 λ 1 , λ 2 , λ 3 A 8 = tr( A ) = λ 1 + λ 2 + λ 3 50 = det( A ) = λ 1 λ 2 λ 3 λ 1 = z C \ R λ 2 = z λ 3 = 2 8 = 2 + 2Re( z ) 25 = k z k 2 Re( z ) = 3 k z k 2 = 25 Im( z ) = 4 3 ± 4 i n 2 T : R n R n T x 1 x 2 x n = x 1 + · · · + x n x 1 + · · · + x n x 1 + · · · + x n T V T : V V T ( a 0 , a 1 , a 2 , a 3 , . . . ) = ( a 1 , a 2 , a 3 , . . . ) T T A A - λI n = 1 - λ 1 · · · 1 1 1 - λ · · · 1 1 1 · · · 1 - λ R i R i R i - R 1 R 1 1 - λ 1 · · · 1 λ - λ · · · 0 λ 0 · · · - λ .