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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.

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    @Arturo Magidin, Thank you, exactly right2012-02-23

1 Answers 1

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In the absence of additional assumptions about the matrix (such as: symmetric, triangular, orthogonal, etc), the matrix form does not really help. The problem is exactly the same as for the list of $mn$ integer values. If each of these integers takes $k$ bits to store, the similarity function $f$ could be their concatenation into a string of $mnk$ bits. This is essentially the function given by Robert Israel: $f(M)=\sum_{ij} a_{ij}B^{i+mj}$ where $B$ is an upper bound on the entries of $M$.

If more is known about the matrix, then some compression is possible: for example, a square antisymmetric matrix of size $n$ is determined by its $n(n-1)/2$ entries lying above the main diagonal.