I am currently studying a robotics paper, where a normalized frobenius norm of a matrix is calculated as part of determining the condition number of a corresponding linear system of equations. The condition number will later be utilized as an objective function for a design optimzation algorithm.
$\mathbf{W}$ is defined as $$\mathbf{W} = \frac{1}{n} \mathbf{I}$$ and inserting this definition into $\left|\left|\mathbf{J}\right|\right|$ results in $$\left|\left|\mathbf{J}\right|\right| = \sqrt{\text{tr}\left(\mathbf{J}\mathbf{W}\mathbf{J}^{T}\right)} =\sqrt{\text{tr}\left(\mathbf{J}\frac{1}{n}\mathbf{I}\mathbf{J}^{T}\right)} = \frac{1}{n}\sqrt{\text{tr}\left(\mathbf{J}\mathbf{I}\mathbf{J}^{T}\right)} = \frac{1}{n}\sqrt{\text{tr}\left(\mathbf{J}\mathbf{J}^{T}\right)} .$$
The normalization factor is therefore independet of $\mathbf{J}$. Therefore, what is the purpose of scaling $\left|\left|\mathbf{J}\right|\right|$ down by a factor proportional to its dimension?
In terms using $\kappa(\mathbf{J})$ to optimize a mechanism, this would mean that the respective value of each iteration would simply be scaled down by constant factor, and therefore not influence the results. Am I missing something significant here?
Thanks in advance!