Consider a sparse $N \times N$ matrix $P$ with at most $2N$ nonzero entries and $\text{diag}(P) = 1$. Define the norm $|P|_1$ as the sum of the absolute values of all entries in $P$, i.e., $|P|_1=\sum_{i, j}\left|P_{i j}\right|$.
For which matrices $P$ does the inequality $|UP|_1 \geq |P|_1$ hold for every unitary matrix $U$, while ensuring that $\text{diag}(UP) = 1$?
Here, $|UP|_1$ is calculated as $\sum_{i, j}\left|(UP)_{i j}\right|$, where $(UP)_{i j}$ is the $(i, j)$ entry of the matrix product $UP$, and can also be expressed as $\sum_{i, j}\left|u_i^T p^j\right|$. (For a matrix $X$, let $x_i$ and $x^j$ denote the $i$-th row and $j$-th column of $X$, respectively.)
Background:
In my research, I developed an algorithm to estimate $P$, and it generally converges correctly (I'm sorry details and code cannot be shared). And if my algorithm can find the correct solution, this problem is a corollary. The algorithm's effectiveness suggests that there might be a broad class of matrices $P$ for which this condition holds.
Empirically, when $P$ is initialized with $N$ off-diagnal elements randomly chosen within the range from -1 to 1 , convergence is fast. However, when the elements are all set to 1 , convergence typically fails. I'm looking to understand what specific properties or characteristics of $P$ ensure the condition $\|U P\|_1 \geq\|P|_1$ is always met.
Update:
Considering a simplified scenario where all entries of $P$ are positive might provide a useful starting point. When analyzing $\|UP\|_1 \geq \|UP_1\|_1 = \|Up\|_1$, with $P_1$ defined as a column vector of all ones, we examine $\|Up\|_1$ by evaluating the sum of the magnitudes of the projections of $U$'s row vectors onto $p$. Here, $p = P_{1,1}$ can be any column vector, and we express:
$$ \|Up\|_1 = \sum_i |u_i p| = \sum_i \|u_i\|_2 \|p\|_2 \cos \theta_i, $$
where $\cos \theta_i$ is the angle between $p$ and $u_i$. This formulation simplifies to the minimum at $\|p\|_2$, but unfortunately, $\|p\|_2$ is not necessarily greater than $\|P\|_1$. This observation suggests that the initial diagonal and sparsity constraints might need to be revisited or revised in the theoretical model.
Update 8.2
To confirm the correctness of the problem, I have developed a program. Dealing with the constraint of unitarity was challenging. My approach involved projecting a non-unitary matrix onto the space of unitary matrices, which is an application of the Orthogonal Procrustes problem.
The code is iterative, ensuring that matrix $U$ ultimately remains unitary while also maintaining that $\operatorname{diag}(UP) = 1$.
For those interested in reviewing or utilizing the methodology, the code can be accessed through the following link: Code.
I welcome any feedback or suggestions on the implementation and the theoretical approach.
