Re: [理工] [工數]-對角化
※ 引述《magiciano (magiciano)》之銘言：
: 出自zill Ch8.12 430頁 習題34
: identify the given conic section. Graph
: 16x^2+24xy+9y^2-3x+4y=0
: 想請問-3x+4y那邊要怎麼處理?...
令
Q=16x^2+24xy+9y^2-3x+4y=
X=[x y]
16 12 x x T
=[x y] [ ] [ ]+ [-3 4] [ ]= X A X + BX
12 9 y y
由det(A-入I)=0 , 可得 入=0 , 25
1 3 1 4
入=0, V1=--- [ ] 入=25, V2=--- [ ]
5 -4 5 3
1 3 4 T -1
令S=[v1 v2]= --- [ ] , S =S
5 -4 3
-1 0 0 u
D=S AS = [ ], 再令 X=SY, Y=[ ]
0 25 v
T T
帶回 X A X + BX =(SY) A (SY) + B(SY)
T T T
=Y S ASY + BSY = Y D Y +BSY
0 0 u 1 3 4 u
=[u v] [ ] [ ] + [-3 4]*---*[ ] [ ]
0 25 v 5 -4 3 v
2 2
=25v - 5u =0 => 5 v - u =0
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