Because exact preconditioning via full eigendecomposition is computationally prohibitive, a significantly cheaper alternative is to approximate the distortion by rescaling the problem using only the diagonal entries of the matrix $\mathbf{Q}$. This diagonal preconditioning calculates a new matrix $\tilde{\mathbf{Q}} = \textrm{diag}^{-\frac{1}{2}}(\mathbf{Q}) \mathbf{Q} \textrm{diag}^{-\frac{1}{2}}(\mathbf{Q})$. In this rescaled representation, the entries become $\tilde{\mathbf{Q}}_{ij} = \mathbf{Q}_{ij} / \sqrt{\mathbf{Q}_{ii} \mathbf{Q}_{jj}}$, ensuring that every diagonal element $\tilde{\mathbf{Q}}_{ii} = 1$. In many scenarios, particularly when the problem is roughly axis-aligned, this straightforward rescaling considerably reduces the condition number without the massive cost of computing true eigenvalues.