Chatbot math/ChatGPT/24.02/Unitary Transformation & Matrix Symmetry


 * Compare the same question on:    ChatGPT    Bard    Copilot

Wow! ChatGPT 3.5 (the free version) nailed the question. One downside is that pasting in Edit mode loses much of the formatting. RTTR on this page looks like $$R^T T R$$ on my computer screen.

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Here is the chat.
- Question: If you subjet a symmetric tensor to a rotation, do the eigenvalues change?

Spelling is correct on subjet (should have been subject) - The behavior of eigenvalues under rotation depends on the type of tensor and the nature of the rotation. For symmetric tensors, it's helpful to consider how the eigenvalues transform under orthogonal transformations, such as rotations.

If you have a symmetric tensor T, and you subject it to a rotation R, the eigenvalues of the rotated tensor RTTR will remain the same as the eigenvalues of the original tensor T. This is because orthogonal transformations, like rotations, preserve the lengths of vectors, and the eigenvalues of a symmetric tensor are related to the principal axes lengths.

Mathematically, if v is an eigenvector of the original tensor T corresponding to eigenvalue λ, then RTTRv will be an eigenvector of the rotated tensor corresponding to the same eigenvalue λ.

In summary, for symmetric tensors, eigenvalues are invariant under orthogonal transformations such as rotations.

To find the eigenvalues of the given 2x2 matrix, we need to solve the characteristic equation, which is obtained by subtracting the eigenvalue λ times the identity matrix from the given matrix and then taking the determinant.

-  Question: Find the eigenvalues of $$\begin{matrix} 2 & 1 \\ 1 & 4 \end{matrix}$$  -

The matrix is:

$$\begin{bmatrix} 2 & 1 \\ 1 & 4 \end{bmatrix} -\lambda I$$

where \(I\) is the 2x2 identity matrix) from \(A\): \[ A - \lambda I = \begin{bmatrix} 2 - \lambda & 1 \\ 1 & 4 - \lambda \end{bmatrix} \]

Now, calculate the determinant: $$\text{det}(A - \lambda I) = (2 - \lambda)(4 - \lambda) - (1 \cdot 1) $$

\] Expand and simplify: \[ \text{det}(A - \lambda I) = \lambda^2 - 6\lambda + 7 \] Now, set the determinant equal to zero and solve for \(\lambda\): \[ \lambda^2 - 6\lambda + 7 = 0 \] This quadratic equation factors as: \[ (\lambda - 1)(\lambda - 7) = 0 \]

'''So, the eigenvalues (\(\lambda\)) are 1 and 7. '''

This is the correct answer. Unfortunately it does not easily render on my screen or when pasted into wikitext.

But ChatGPT is better than Copilot which is better than Bard on this task.