Re: [線代] 請教一題主軸定理

[LiamIssac (Madchester)]

※ 引述《akiu072 (Bin)》之銘言：
: 設Q(X)=x^2 + 2y^2 + 2yz + 2z^2 = 1
: .
: .
: .
: 在計算主軸定理時，依照課本做法都會先拆成一個矩陣
: 然後再求特徵值、特徵向量...
: 後續的做法我應該沒問題，但要怎麼把上述的Q(X)拆成能相乘的矩陣

Since your Q(X) is quadratic, it can be written as

           Q(X) = X^{T}EX

where X = [x, y, z]^{T} is a 3-D column vector and

      E is the coeff matrix you need

Then, according to your Q(X)

      E = [1 0 0
           0 2 1
           0 1 2]


Easy way to construct E

E_{1,1} correponds to x^2

E_{2,2}            to y^2

E_{3,3}            to z^2

so 1 means x, 2 means y, and 3 means z

Then the yz in Q(X) corresponds to E_{2,3} (and of course E_{3,2})


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