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Suppose I have a matrix $A = [a_{ij}]_{n\times n}$ such that $a_{ij} \in \mathbb{N}$ and $a_{ij}=a_{ji}$. I want to transform this matrix such that the sum $\sum_{i,j}a_{ij}$ remains constant, symmetry is maintained, and the resultant matrix also comprises natural numbers. Identity transformation is one example of such a transformation.

If we define the operator $\sum$ as $\sum A = \sum_{i,j}a_{ij}$, then $\sum (AT) = \sum A$ or $\sum (TA) = \sum A$.

Examples

  1. Identity transform.
  2. Subtract a number less than or equal to $a_{ij}$ or $a_{ji}$ from $a_{ij}$ and $a_{ji}$ and add it to $a_{kl}$ and $a_{lk}$ for arbitrary indices.
  3. Swapping of diagonal elements: $a_{ii}\to a_{jj}$ and $a_{jj}\to a_{ii}$.

I am wondering if there is a general formulation for such transformations, possibly in matrix form. And could someone provide insight into left and right transforms?

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