Answer to Question #125838 in Linear Algebra for hitendra GARG

Question

Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1 = {(1, 0, 0),(1, 1, 0),(0, 0, 1)},B2 = {(1, 0, 0),(0, 1, 2),(0, 2, 1)}. If Q(x) = x1^2-4x1x2+2x2x3+x2^2+x3^2 , find the representation of Q in terms of (y1,y2,y3)

Answer 1

Let $e_{1} = eS_1, e_2 = eS_2, e_2 = e_1S$$, e = {(1;0;0), (0;1;0), (0;0;1)}$ and $A = \begin{pmatrix} 1& -2 & 0\\ -2& 1 & 1\\ 0 & 1 & 1 \end{pmatrix}$ is the mathix of Q(x).

Then $S_1 = \begin{pmatrix} 1& 1 & 0\\ 0& 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$ , $S_2 = \begin{pmatrix} 1& 0 & 0\\ 0& 1 & 2\\ 0 & 2 & 1 \end{pmatrix}$ , $e = e_1S_1^{-1}$ , $e_2 = eS_2 = e_1S_1^{-1}S_2$

That is $S = S_1^{-1}S_2$

$S_1^{-1} = \begin{pmatrix} 1& -1 & 0\\ 0& 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$ $S =S_1^{-1}S_2 = \begin{pmatrix} 1& -1 & 0\\ 0& 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1& 0 & 0\\ 0& 1 & 2\\ 0 & 2 & 1 \end{pmatrix} = \begin{pmatrix} 1& -1 & -2\\ 0& 1 & 2\\ 0 & 2 & 1 \end{pmatrix}$

$A' = S^TAS = \begin{pmatrix} 1& 0 & 0\\ -1& 1 & 2\\ -2 & 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1& -2 & 0\\ -2& 1 & 1\\ 0 & 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1& -1 & -2\\ 0& 1 & 2\\ 0 & 2 & 1 \end{pmatrix}$ $=$

$= \begin{pmatrix} 1& -2 & 0\\ -3& 5 & 3\\ -6 & 7 & 3 \end{pmatrix} \cdot \begin{pmatrix} 1& -1 & -2\\ 0& 1 & 2\\ 0 & 2 & 1 \end{pmatrix}$ $= \begin{pmatrix} 1& -3 & -6\\ -3& 14 & 19\\ -6 & 19 & 29 \end{pmatrix}$ .

So, $Q(y) = y_1^2 - 6y_1y_2 -12y_1y_3 + 14y_2^2 + 38y_2y_3 +29y_3^2$