I've read from a book that "An ill-conditioned matrix can be transformed to a well-conditioned one through Lattice reduction algorithms."
The ill-conditioned matrix means that the condition number is large, while the well-conditioned matrix refers to a small condition number.
I've browsed through the internet, but I did not find any information on this.
I would like to ask if this is true?
If this is true, how can it be done?
More formally, given an ill-conditioned square matrix $A\in{\mathbf{C}^{N\times{N}}}$, is it possible to find a lattice reduction matrix $P$ such that $AP$ is well-conditioned? How can we find the matrix $P$?
I would like to know this property since I have got a bunch of ill-conditioned matrices and intend to solve the linear system by some convex optimization techniques after the lattice transform (ex. Jacobi-Method, Coordinate descent, ...).