Is anyone aware of a orthogonal multiple regression library that is implemented in say R, Scipy, Matlab, Octave, etc.? (Or even fortran/C...) If I'm not mistaken, it would not be difficult to write but just wanted to check.
Second question: my actual application is for a multivariate regression case where I have two matrices, $\mathbf{Y}$ and $\mathbf{X}$. If I am able to find a total least squares estimator for $\boldsymbol\beta$ and $\boldsymbol\beta'$, would the inverse of its transposed elements equal one another ($\hat{\beta}_{ji}^{-1} = \hat{\beta}'_{ij}$) if the two coefficient matrices are defined by $\mathbf{Y} = \hat{\boldsymbol{\beta}} \mathbf{X}$ and $\mathbf{X} = \hat{\boldsymbol{\beta}}' \mathbf{Y}$? It seems that this equality ($\hat{\beta}_{ji}^{-1} = \hat{\beta}'_{ij}$) does not necessarily hold as it does for the classic case, where the scalar values of $\beta^{-1}=\beta'$ when $\mathbf{Y}$ and $\mathbf{X}$ are simply vectors rather than matrices, but also wanted to check if this was a known fact.
Thanks.