I wanted to conduct a total least squares regression on two variables. My statistical programme does not provide TLS, but TLS luckily equals Principal Component Analysis, as far as I know. Since all variables are standardized, this is done by applying a SVD on the correlation matrix of the concerned variates.
Now, my take on the issue was to read out the first eigenvector to obtain the TLS coefficients. However, the SVD gives me the same eigenvector (weights) irrespective of what the two variables are. It's always
[.70710678, .70710678]. I find this strange. Of course, the eigenvalues differ.
My questions are: How to interpret this? Has this result maybe anything to do with the coordinate space, like it is just a matter of rotation? One note: I used Stata to apply the SVD. To be clear: The question is not about TLS directly, but why I get the same eigenvectors irrespective of which variables I use (as long as they are exactly 2).