First time asker here.
First off, I know I should be doing this in R or Python. I will. For now I'm reading a textbook, using simple examples and Excel to try to learn the concepts of linear algebra. I've got an Excel add-in from Ponderosa Computing. It displays eigenvalues and eigenvectors in ascending order, and singular values and vectors in descending order. The book says right singular vectors are the eigenvectors of ATA. I think I understand that. Here is my matrix $A$, which is a centered difference matrix (centered by column), the matrix ATA$/(n-1)$, and eigenvectors of ATA:
Now here's what's getting me confused:
The right singular vectors of the difference matrix $A$ are the eigenvectors of ATA but with signs reversed. The add-in automatically displays the vs transposed, ostensibly to make the SVD math easier to see.
I transpose back, then multiply by an Identity matrix with rows switched in order to display the singular vectors in the same way as the above eigenvectors.
So, what's the interpretation here? I've read over and over again that right singular vectors (vs) are just the eigenvectors of ATA. But these are different vectors. I remember reading about having to change the sign of the determinant when matrices are acted on by permutation matrices (or something). Is that what's going on? Are the eigenvectors in my third image and singular vectors in my last image equally valid?
Thanks if you read this whole thing. Struggling here