I was trying to solve a linear system:
$$ \mathbf{A}\mathbf{x} = \mathbf{y} $$
but the conditioning number was quite bad (around $10^{17}$). I thought that the system was singular, but after scaling the system using the diagonal matrix:
$$ \mathbf{D_{ii}} = 1/\sqrt{\mathbf{A_{ii}}} $$
and the fact:
$$ \mathbf{A}^{-1} = \mathbf{D} (\mathbf{D}\mathbf{A}\mathbf{D})^{-1}\mathbf{D}. $$
The conditioning number of $\mathbf{D}\mathbf{A}\mathbf{D}$ is around $10^3$! Does this imply that the system was just badly scaled and that there is no collinearity?
Thanks!