Conic Equation using a Quadratic Form

[Apple&Orange]

Homework Statement

x₁²+x₁x₂+2x₂²=8

a) Write the equation using a quadratic form i.e. $\underline{x}$^TA$\underline{X}$=8

b)Find the Matrix Q such that the transformation $\underline{X}$=Q$\underline{Y}$ diagonalises A and reduces the quadratic form to standard form in terms of coordinates (y₁,y₂)

Homework Equations

$\underline{X}$=Q$\underline{Y}$
$\underline{X}$^TA$\underline{X}$=8

The Attempt at a Solution

For question b), I got the A matrix as [1 1;0 2] or [1 0.5;0.5 2] *sorry, don't know how to use the matrix operator so I've written it MATLAB style*.

I used the first matrix to give a better looking eigenvalues, which resulted in 2 and 1. From the values, I got a vector of [1;0] and [1;1]

Using the vectors, I got a Q matrix of [1 0; 1/sqrt(2) 1/sqrt(2)]
and using $\underline{X}$=Q$\underline{Y}$, I got
2.707y₁²+2.707y₁y₂+y₂² which I'm not even sure if its right.

Could someone please assist me in tackling this question?
Thanks!

 

[HallsofIvy]

Yes, the matrix here is
$$\begin{bmatrix}1 & 0.5 \\ 0.5 & 2\end{bmatrix}$$

The first matrix you give is wrong. In order to be certain that there are eigenvalues, you must have a symmetric matrix. You don't want "better looking" eigenvalues, you want the right eigenvalues!

Finally, the problem asked you to "Find the Matrix Q such that the transformation X=QY diagonalises A reduces the quadratic form to standard form in terms of coordinates (y1,y2)" but your final result is NOT in standard form.

 

[Apple&Orange]

Yes, the matrix here is
$$\begin{bmatrix}1 & 0.5 \\ 0.5 & 2\end{bmatrix}$$

The first matrix you give is wrong. In order to be certain that there are eigenvalues, you must have a symmetric matrix. You don't want "better looking" eigenvalues, you want the right eigenvalues!

Finally, the problem asked you to "Find the Matrix Q such that the transformation X=QY diagonalises A reduces the quadratic form to standard form in terms of coordinates (y1,y2)" but your final result is NOT in standard form.

Got it! Thanks!