Given a matrix:
$$ A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix} $$
$a_{i,j}$ is a signless integral and bounded. And $b_{i,j}$ is the same. Is there any similarity function between $a_{i,j}$ and $b_{i,j}$. Such that $f(a_{i,j})=f(b_{i,j})$ if and only if $a_{i,j}=b_{i,j}$.
For example, the rank of matrices can identify a class of matrices, not a single matrix.
