vela said:Let
[tex]\vec{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}[/tex]
Find the matrix A such that
[tex]\vec{x}^\mathrm{T} A \vec{x} = 2x_1^2+4x_2^2+5x_3^2-4x_1x_3[/tex]
You want to diagonalize this matrix.
vela said:Try a 2x2 example. Calculate
[tex]\begin{pmatrix} x & y \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}[/tex]
Compare it to [itex]Ax^2 + Bxy + Cy^2[/itex]. 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?
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.
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.
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.
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.
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.