Dodging to do conjugate gradient on the normal equations.

Let us consider the linear equation system $ $ \bf Ax = b$ $

We can formulate it’s normal equations:

$ $ {\bf A}^T{\bf Ax=A}^T{\bf b}$ $

but these are often harder to solve, because $ {\bf A}^T{\bf A}$ has worse condition number than $ \bf A$ . Some time ago I stumbled upon this approach to formulate a linear equation system without normal equations:

$ $ \begin{bmatrix}\bf I&\bf A\{\bf A}^T&\bf 0\end{bmatrix}\begin{bmatrix}\bf r\\bf x\end{bmatrix}=\begin{bmatrix}\bf b\\bf 0\end{bmatrix}$ $

Here we can see the constructed matrix is symmetric so we can avoid the normal equations, but at the expense of blowing up the space to dim(r)+dim(x).

How can we prove / realize that this choice will be better/faster?