※ 引述《PttFund (批踢踢基金只進不出)》之銘言：
: Let the matrix
: A = [ 3 1 ].
: [ -1 1 ]
: (a) Find the characteristic polynomial and minimal polynomial
: of A.
: (b) Use the minimal polynomial of A in (a) to express A^100 as
: a linear combination of A and I.
: (c) Use (b) to compute A^100.
(a)
A的特徵多項式 = det(A - xI) = det[ 3-x 1 ]
[ -1 1-x ]
= (3-x)(1-x) + 1 = x^2 - 4x + 4 = (x-2)^2
A - 2I = [ 1 1 ]
[ -1 -1 ]
(A-2I)^2 = [ 1 1 ][ 1 1 ]
[ -1 -1 ][ -1 -1 ]
= [ 0 0 ]
[ 0 0 ]
所以 A 的最小多項式是 (x-2)^2
(b)
令 f(x) = x^100 = (q(x))*(x-2)^2 + a(x-2) + b
x = 2 代入得 b = 2^100
x = 1 代入得 -a + b = 1
所以 a = b - 1 = 2^100 - 1
因此 A^100 = f(A) = (q(A))*(A-2I)^2 + a(A-2I) + bI
= a(A-2I) + bI
= (2^100 - 1)*(A-2I) + (2^100)*I
= (2^100 - 1)*A + (-2^100 + 2)I
(c)
A^100 = (2^100 - 1)A + (-2^100 + 2)I
= (2^100 - 1)[ 3 1 ] + (-2^100 + 2)[ 1 0 ]
[ -1 1 ] [ 0 1 ]
= [ 2^101 - 1 2^100 - 1 ]
[ -2^100 + 1 1 ]
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