# Constraining the element-wise magnitudes of a matrix

• weetabixharry
In summary, the goal is to find a unitary matrix, \textbf{U} which minimizes "something like"\underline{1}^T \left|\left(\left|\textbf{B}_{opt}\right|^2 - \underline{1}\underline{1}^T\right)\right|^2subject to \textbf{U}^H\textbf{U}=\textbf{I}. However, differentiating the constraint term produces strange results. The author has been using the Frobenius norm to write the matrix trace, but has run into trouble with the term not being summed. They are looking for any fresh

#### weetabixharry

I have a complex matrix, $\textbf{A}$, and I want to left-multiply it by a unitary matrix, $\textbf{U}$ (i.e. $\textbf{U}$ is square and $\textbf{U}^H\textbf{U}=\textbf{I}$).

The goal is to find the $\textbf{U}$ which yields the optimal solution $\textbf{B}_{opt} \triangleq \textbf{U}\textbf{A}$, where $\textbf{B}_{opt}$ is optimal in the sense that its element-wise magnitudes are all simultaneously as close as possible to unity.

That is, I would like to minimise something like: $\underline{1}^T \left|\left(\left|\textbf{B}_{opt}\right|^2 - \underline{1}\underline{1}^T\right)\right|^2 \underline{1}$ (subject to $\textbf{U}^H\textbf{U}=\textbf{I}$), where $\left|\textbf{B}_{opt}\right|$ denotes the element-by-element absolute value of $\textbf{B}_{opt}$ and $\underline{1}$ is a column vector of ones.

How can I approach this problem? I have tried to find a solution using Lagrange multipliers, but I can't seem to gain any insight into how to design $\textbf{U}$. What other sorts of methods are available for this type of problem?

The "element-by-element magnitude squared" can be written

$$\mathrm{tr} \, (A^\dagger A)$$

This might be useful.

Ben Niehoff said:
The "element-by-element magnitude squared" can be written

$$\mathrm{tr} \, (A^\dagger A)$$

This might be useful.

Ah yes - I have been using that expression (sum of element-by-element magnitude squared is the Frobenius norm). It is particularly useful because matrix traces tend to be easy to differentiate.

However, I run into trouble with the $\left|\textbf{B}_{opt}\right|^2$ term, which is not summed. I wrote it as $\left( \textbf{B}_{opt} \odot \textbf{B}_{opt}^* \right)$, where $\odot$ is the Hadamard (element-by-element) product. When differentiated (w.r.t. $\textbf{U}^*$) this produces an ugly expression with several Hadamard products that I find difficult to work with.

Furthermore, differentiating the constraint term in the Lagrange function seems to give strange results (or perhaps I have made an error)... so I'm looking for any fresh ideas or insights!