Defactoring algorithms: Difference between revisions

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The HNF is used out of convenience, but in fact the conditions necessary to achieve defactoring are less strict. This was pointed out by Tom Price when he explained his defactoring method.

The conditions for attaining the HNF are strict enough to enforce uniqueness, in particular the condition that entries in pivot columns must be between 0 and the pivot. This part is not relevant to defactoring. The only thing that is critical is that all-zero columns are created such that all the information in the matrix has been compressed into a single square submatrix, i.e. into a single non-zero minor determinant, and of course that this transformation has been achieved using only unimodular operations, so that the image of the matrix remains the same.

The conditions for attaining the HNF are strict enough to enforce uniqueness, in particular the condition that entries in pivot columns must be between 0 and the pivot. This part is not relevant to defactoring. The only thing that is critical is that all-zero columns are created such that all the information in the matrix has been compressed into a single square submatrix, i.e. into a single non-zero minor determinant, and of course that this transformation has been achieved using only unimodular operations, so that the [[image]] of the matrix remains the same.

So fitting the information from a rectangle into a square is the target, although in practice this is usually only possible by getting the information into echelon form.

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 The HNF is used out of convenience, but in fact the conditions necessary to achieve defactoring are less strict. This was pointed out by Tom Price when he explained his defactoring method.
-The conditions for attaining the HNF are strict enough to enforce uniqueness, in particular the condition that entries in pivot columns must be between 0 and the pivot. This part is not relevant to defactoring. The only thing that is critical is that all-zero columns are created such that all the information in the matrix has been compressed into a single square submatrix, i.e. into a single non-zero minor determinant, and of course that this transformation has been achieved using only unimodular operations, so that the image of the matrix remains the same.
+The conditions for attaining the HNF are strict enough to enforce uniqueness, in particular the condition that entries in pivot columns must be between 0 and the pivot. This part is not relevant to defactoring. The only thing that is critical is that all-zero columns are created such that all the information in the matrix has been compressed into a single square submatrix, i.e. into a single non-zero minor determinant, and of course that this transformation has been achieved using only unimodular operations, so that the [[image]] of the matrix remains the same.
 So fitting the information from a rectangle into a square is the target, although in practice this is usually only possible by getting the information into echelon form.