Reduce the quadratic form 3x12+3x22+3x32+2x1x2+2x1x3−2x2x3 into 'a sum of squares' by an orthogonal transformation and give the matrix of transformation.
Solution:
On comparing the given quadratic with the general quadratic ax2+by2+cz2+2fyz+2gzx+2hxy , the matrix is given by
A=⎣⎡ahghbfgfc⎦⎤=⎣⎡31113−11−13⎦⎤
The desired characteristic equation becomes
∣A−λI∣=∣∣3−λ1113−λ−11−13−λ∣∣=0
which is a cubic in λ and has three values viz., 1,4,4.
Hence the desired canonical form i.e., 'a sum of squares' is x2+4y2+4z2 . Solving [A−λI][X]=0 for three values of λ
For λ=1 , we have ⎣⎡21112−11−12⎦⎤⎣⎡x1y1z1⎦⎤=0or2x1+y1+z1=0x1+2y1−z1=0},i.e.−1−2x1=1+2y1=4−1z1=k∴
⎣⎡x1y1z1⎦⎤=⎣⎡−kkk⎦⎤=⎣⎡−111⎦⎤
Similarly for λ=4,⎣⎡−1111−1−11−1−1⎦⎤⎣⎡xyz⎦⎤=0
We have two linearly independent vectors X2=⎣⎡110⎦⎤,X3=⎣⎡101⎦⎤
As the transformation has to be an orthogonal one, therefore to obtain ' P ', first divide each elements of a corresponding eigen vector by the square root of sum of the squares of its respective elements and then express as [X Y Z]
Hence the matrix of transformation, P=⎣⎡3131312121021021⎦⎤
Comments