Reduce the quadratic form Q(x,y,z)=x2+5y2+z2+2xy+2yz+6xz to canonical form.
The given quadratic form is
Q(x,y,z)=x2+5y2+z2+2xy+2yz+6xz The matrix of the given quadratic form is
A=⎝⎛113151311⎠⎞
A−λI=⎝⎛1−λ1315−λ1311−λ⎠⎞
det(A−λI)=∣∣1−λ1315−λ1311−λ∣∣
=(1−λ)∣∣5−λ111−λ∣∣−∣∣1311−λ∣∣+3∣∣135−λ1∣∣
=(1−λ)(5−5λ−λ+λ2−1)−(1−λ−3)
+3(1−15+3λ)=4−6λ+λ2−4λ+6λ2−λ3
+2+λ−42+9λ
=−λ3+7λ2−36=0
−λ2(λ−6)+(λ−6)(λ+6)=0
−(λ−6)(λ2−λ−6)=0
−(λ−6)λ−3)(λ+2)=0
λ1=6,λ=3,λ3=−2These are the eigenvalues.
λ=6
⎝⎛1−λ1315−λ1311−λ⎠⎞=⎝⎛−5131−1131−5⎠⎞
⎝⎛−5131−1131−5⎠⎞→⎝⎛100010−1−20⎠⎞ The eigenvector is v=⎝⎛121⎠⎞
λ=3
⎝⎛1−λ1315−λ1311−λ⎠⎞=⎝⎛−21312131−2⎠⎞
⎝⎛−21312131−2⎠⎞→⎝⎛100010−110⎠⎞ The eigenvector is u=⎝⎛1−11⎠⎞
λ=−2
⎝⎛1−λ1315−λ1311−λ⎠⎞=⎝⎛313171313⎠⎞
⎝⎛313171313⎠⎞→⎝⎛100010100⎠⎞ The eigenvector is w=⎝⎛−101⎠⎞
Form the matrix P, whose column i is eigenvector no. i
P=⎝⎛1211−11−101⎠⎞Form the diagonal matrix D, whose element at row i, column i is eigenvalue no. i
D=⎝⎛60003000−2⎠⎞The matrices P and D are such that
P−1AP=D Hence the quadratic form is reduced to a sum of squeres, i.e. canonical form:
6x~2+3y~2−2z~2P=⎝⎛1211−11−101⎠⎞ is the matrix of transformation which is an orthogonal matrix.
The canonical form of the quadratic form is
6x~2+3y~2−2z~2
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