- #1

- 111

- 0

## Main Question or Discussion Point

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

The goal is to find the [itex]\textbf{U}[/itex] which yields the optimal solution [itex]\textbf{B}_{opt} \triangleq \textbf{U}\textbf{A}[/itex], where [itex]\textbf{B}_{opt}[/itex] 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: [itex]\underline{1}^T \left|\left(\left|\textbf{B}_{opt}\right|^2 - \underline{1}\underline{1}^T\right)\right|^2 \underline{1}[/itex] (subject to [itex]\textbf{U}^H\textbf{U}=\textbf{I}[/itex]), where [itex] \left|\textbf{B}_{opt}\right|[/itex] denotes the element-by-element absolute value of [itex]\textbf{B}_{opt}[/itex] and [itex]\underline{1}[/itex] 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 [itex]\textbf{U}[/itex]. What other sorts of methods are available for this type of problem?

Any advice is greatly appreciated!

The goal is to find the [itex]\textbf{U}[/itex] which yields the optimal solution [itex]\textbf{B}_{opt} \triangleq \textbf{U}\textbf{A}[/itex], where [itex]\textbf{B}_{opt}[/itex] 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: [itex]\underline{1}^T \left|\left(\left|\textbf{B}_{opt}\right|^2 - \underline{1}\underline{1}^T\right)\right|^2 \underline{1}[/itex] (subject to [itex]\textbf{U}^H\textbf{U}=\textbf{I}[/itex]), where [itex] \left|\textbf{B}_{opt}\right|[/itex] denotes the element-by-element absolute value of [itex]\textbf{B}_{opt}[/itex] and [itex]\underline{1}[/itex] 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 [itex]\textbf{U}[/itex]. What other sorts of methods are available for this type of problem?

Any advice is greatly appreciated!