I have an engineering problem in which I use SVD on matrix $\mathbf{A}$:
\begin{align} \mathbf{A} &= \textbf{U} \mathbf{\Sigma} \textbf{V}^{*} \end{align}
However, due to the fact that the singular values in the diagonal of $\mathbf{\Sigma}$ have a fairly large range, in my application some rows are more benefited than the others. Without going deep into the application (I know I`m not explaining it), it suffices to understand a large magnitude of the value in each row of $\mathbf{\Sigma} $ is desirable. Now, what I wanted to have was a form of decomposing $\mathbf{A}$ into a product "UnitaryMatrix x Diagonal x UnitaryMatrix", but such that the magnitudes in the diagonal of the diagonal matrix were more evenly distributed.
Here is a sample of the diagonal matrix in my application: \begin{align} \mathbf{\Sigma} = \left[ \begin{array}{ l l l l } 1.3333&0&0&0\\ 0&0.84117&0&0\\ 0&0&0.54855&0\\ 0&0&0&0.4738\\ \end{array} \right] \end{align}
It would be better, for example, to have \begin{align} \mathbf{\Sigma} = \left[ \begin{array}{ l l l l } 0.8&0&0&0\\ 0&0.8&0&0\\ 0&0&0.8&0\\ 0&0&0&0.8\\ \end{array} \right] \end{align}
Note, however, I can't simply redefine the diagonal matrix, because $\mathbf{A}$ is determined by physical properties which can`t be modified. Thus, the question is, is there a decomposition that would serve for this purpose (diagonal with more evenly distributed magnitudes)?
Given the singular value decomposition $$ \mathbf{A} = \mathbf{U} \, \Sigma \, \mathbf{V}^{*} $$ we note that the singular values are unique. The spectrum of singular values typically spans many orders of magnitude. They represent scale factors between the aligned orthonormal bases for the domain and the codomain. When you change the singular values, you change the scaling between the range spaces. You change the output matrix $\mathbf{A}$.
The following shows an example of singular value decomposition homogenization. Start with a 256 $\times$ 256 matrix, an image of Camille Jordan, scaled such that all values are within $[0,1]$. The image and its spectrum are shown below.
Next, the singular values are all set to unity, $$ \Sigma = \mathbf{I}_{256}, $$ and the image is reassembled. The range of data values is $[0.396485,-0.383183]$.