Answer to Question #139232 in Linear Algebra for Varun

Question

Reduce the quadratic form x1^2 + 2X2x3 to canonical form

Answer 1

Given quadratic form is $x_1^2+2x_2x_3$

compare the given quadratic expression with standard quadratic form,

$ax_1^2+by_1^2+cz_1^2+2fyz+2gzx+2hxy$

we get,

a=1,b=0,c=0,f=1,g=0 and h=0

Transforming the quadratic equation into matrix form

So the required matrix is ,

$A=\begin{bmatrix} a & h &g\\ h& b&f\\ g&f&c \end{bmatrix}$ , $A=\begin{bmatrix} 1 & 0 &0\\ 0& 0&1\\ 0&1&0 \end{bmatrix}$

Using characterstics Equation $(A-\lambda I)X=0$

To calculate the values of $\lambda$

$|A-\lambda I|=0$

$\begin{vmatrix} 1 & 0 &0\\ 0& 0&1\\ 0&1&0 \end{vmatrix}-\lambda \begin{vmatrix} 1& 0 &0\\ 0& 1&0\\ 0&0&1 \end{vmatrix}=0$

$\begin{vmatrix} 1-\lambda & 0 &0\\ 0& -\lambda&1\\ 0&1&-\lambda \end{vmatrix}$ =0

Using Factorization method

$(1-\lambda)(\lambda^2-1)=0$

so $\lambda=1,-1,1$ (these are the required eigen values)

S0 the required canonical form is

$\lambda_1x^2+\lambda_2y^2+\lambda_3z^2=0$

$x^2-y^2+z^2=0$

This is the required canonical form.