I have a problem where I am trying to do normal multiple linear regression and estimate some parameters. The model is
$\boldsymbol{Y}=X\boldsymbol{\beta} + \boldsymbol{\epsilon}$, $\epsilon \in N(0, I)$
I have 20 $\beta$ parameters to estimate and am using Matlab to solve using fminunc.
Now what I am looking to do is that instead of estimating $\boldsymbol{\beta}$ directly, I would like to multiply $\boldsymbol{\beta}$ by a matrix of constants, A (size 20 by 20), before doing the maximisation.
I would then like to find the new estimates, say $\boldsymbol{\alpha}$ which could then be multiplied by matrix A to get back my estimates for $\boldsymbol{\beta}$.
The problem am having is that this does not seem to work, ie I cannot get back my estimates for $\boldsymbol{\beta}$ by doing this. The matrix A is orthogonal, ie all columns are linearly independent. There are 20 unknowns and 20 equations so why is this not working?