# Finding Linear Transformation that will remove cross product term.

• iqjump123
In summary, the conversation discusses finding a linear transformation from X={x1,x2,x3} to U={u1,u2,u3} in order to remove the cross product term in a quadratic form. The resulting quadratic form in u1,u2,u3 is then written out. The conversation also mentions using a given matrix to solve the problem and generalizing the solution to the 3x3 case.
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.

## 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

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.

vela said:
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.
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?

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?

vela said:
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 :)

## 1. How do I find a linear transformation that will remove the cross product term?

To find a linear transformation that will remove the cross product term, you can use a process called matrix diagonalization. This involves finding the eigenvalues and eigenvectors of the matrix that represents the cross product term, and then using those values to create a diagonal matrix. This diagonal matrix will be the transformation matrix that removes the cross product term.

## 2. Can any linear transformation remove the cross product term?

No, not all linear transformations can remove the cross product term. The transformation must be orthogonal, meaning it preserves angles and distances, in order to fully remove the cross product term.

## 3. Is it possible to remove the cross product term without using a linear transformation?

No, a linear transformation is necessary to remove the cross product term. This is because the cross product term is a result of the vector multiplication, and a linear transformation is needed to manipulate the vectors in a way that cancels out the cross product term.

## 4. Can I use other mathematical methods besides diagonalization to find a linear transformation that removes the cross product term?

Yes, there are other methods that can be used to find a linear transformation that removes the cross product term. These include eigendecomposition, singular value decomposition, and Gram-Schmidt orthogonalization. However, diagonalization is the most commonly used method for this purpose.

## 5. How do I know if the linear transformation I found is the correct one to remove the cross product term?

You can check if the linear transformation is correct by applying it to the original matrix with the cross product term and seeing if the resulting matrix has a zero in the cross product term's position. If it does, then the linear transformation successfully removed the cross product term.

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