I need to write this equation here: X1^2+X1X2+X2^2=8
in the form of X^TAX=8
Once ive done that i nee to be able to find the matrix Q such that the transformation X=QY diagonalises A and reduces the equation to a standard form in terms of (Y1, Y2)
I need to write this equation here: X1^2+X1X2+X2^2=8
in the form of X^TAX=8
Once ive done that i nee to be able to find the matrix Q such that the transformation X=QY diagonalises A and reduces the equation to a standard form in terms of (Y1, Y2)
You may assume A is symmetric so:
$\displaystyle \left[\begin{array}{cc}x_1&x_2 \end{array} \right]$ $\displaystyle \left[\begin{array}{cc}a_{1,1}&a_{1,2}\\a_{1,2}&a_{2,2}\ end{array}\right]$ $\displaystyle \left[\begin{array}{c}x_1\\x_2\end{array}\right]=8$
Now expand the expression on the left and set the coefficient of $\displaystyle x_1^2$ to $\displaystyle 1$, of $\displaystyle x_2^2$ to $\displaystyle 1$ and of $\displaystyle x_1x_2$ also to $\displaystyle 1$ and solve for $\displaystyle a_{1,1}$, $\displaystyle a_{2,2}$ and $\displaystyle a_{2,2}$
CB