Finding Linear Transformation that will remove cross product term.

[iqjump123]

Homework Statement

Find a linear transofmration from X={x1,x2,x3} to U={u1,u2,u3} which will remove the cross product term in the quadratic form of equation 2X₁²+4X₂²+5X₃²-4X₁X₃
and thus write the resulting quadratic form in u1,u2,u3.

Homework Equations

The Attempt at a Solution

No idea at the moment. I presume that the 4x1x3 term is the term that I need to have disappear, but other than that, I can sure use some help. Thanks

 

[vela]

Let
$$\vec{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$$
Find the matrix A such that
$$\vec{x}^\mathrm{T} A \vec{x} = 2x_1^2+4x_2^2+5x_3^2-4x_1x_3$$
You want to diagonalize this matrix.

 

[iqjump123]

Let
$$\vec{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$$
Find the matrix A such that
$$\vec{x}^\mathrm{T} A \vec{x} = 2x_1^2+4x_2^2+5x_3^2-4x_1x_3$$
You want to diagonalize this matrix.

Hello vela.
Thanks for the reply!
The problem actually gave a matrix for a previous part so just diagonalizing it solved the problem fairly easily. However, if there are no given matrix, how would I find the matrix?

 

[vela]

Try a 2x2 example. Calculate
$$\begin{pmatrix} x & y \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$$
Compare it to $Ax^2 + Bxy + Cy^2$. What values would you choose for a, b, c, and d to get the coefficients A, B, and C? Note that you're shooting for a symmetric matrix. Can you generalize this to the 3x3 case?

 

[iqjump123]

Try a 2x2 example. Calculate
$$\begin{pmatrix} x & y \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$$
Compare it to $Ax^2 + Bxy + Cy^2$. What values would you choose for a, b, c, and d to get the coefficients A, B, and C? Note that you're shooting for a symmetric matrix. Can you generalize this to the 3x3 case?

I see. I understood it now! Thanks a lot for your help :)