Hence there's some permutation of $A$ that does not appear in our list of all $RAC$ matrices.īTW, just to close this out: for $1 \times 1$ matrices, the answer is "yes, all permutations can in fact be realized by row and column permutations." I suspect you knew that. permutation index followed by the elements of the row being randomised. So the number of possible results of applying row- and col-permutations to $A$ is smaller than the number of possible permutations of the elements of $A$. of matrix are designated by numeric value called. &< \\īecause $2n \le n^2$ for $n \ge 2$, and factorial is an increasing function on the positive integers. Such a matrix is always row equivalent to an identity. Where $R$ and $C$ each range independently over all $n!$ permutation matrices, we get at most $(n!)^2$ possible results. A permutation matrix is a square matrix obtained from the same size identity matrix by a permutation of rows. If we consider all expressions of the form Permuting the rows of a matrix based on given a given permutation. There are $n!$ row-permutations of $A$ (generated by premultiplication by various permutation matrices), and $n!$ col-permutations of $A$ (generated by post-multiplication by permutation matrices). Then there are $(n^2)!$ distinct permutations of $A$. Suppose the entries in the $n \times n$ matrix $A$ are all distinct. We prove that our algorithm produces matrices with maximum row sums no more than 3/2 - 1/2m times greater than those found by an optimization rule.
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