Re: [理工] [線代]-二次式
※ 引述《gn00618777 (123)》之銘言：
: By using orthogonal transformations,reduce following quadratic forms to
: sums of squares
: 2 2
: a) 2x1 +x2 -4x1x2-4x2x3
2 -2 0 X1
Q= [X1 X2 X3][-2 1 -2 ] [X2 ]
0 -2 0 X3
=X^t A X
特徵值 4 1 -2
當 λ = 4 V1 = c2[2 -2 1]^t = [2/3 -2/3 1/3]^t
λ = 1 V2 = c3[2 1 -2]^t = [2/3 1/3 -2/3]^t
λ = -2 V3 = c1[1 2 2]^t = [1/3 2/3 2/3 ]^t
令S = [v1 v2 v3] S為正交矩陣 S^t = S^-1
X=SY
4 0 0 y1
Q=(SY)^t A SY = Y^tS^tASY =[y1 y2 y3][0 1 0 ][y2 ]
0 0 -2 y3
Q = 4y1^2 +y2^2 -2y3^2
: b) 8x1x3+2x1x4+2x2x3+8x2x4
b矩陣比較大 囧 不過方法還是一樣
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