Defactoring algorithms: Difference between revisions

According to Wolfram<ref>See: https://reference.wolfram.com/language/ref/SmithDecomposition.html</ref>, "the Smith decomposition can often be found by two iterations of the Hermite decomposition". This notion is echoed in the Sage docs<ref>See: https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_integer_dense.html</ref>, which read, "Use Hermite normal form twice to find an invertible matrix whose inverse transforms a matrix with the same row span as self to its saturation," remembered that saturation is the same as defactoring. The reason for the multiple applications of Hermite decomposition to achieve Smith normal form is that you won't necessarily get a diagonal matrix on the first go. But as we know from Smith defactoring, the center matrix of the Smith decomposition — the Smith normal form one — is not the target matrix, so its exact form is not necessary to achieve to accomplish defactoring.  

Another thought that might help congeal the notion of column Hermite defactoring for you is to use what you know about multimaps (AKA "wedgies"), in particular a) what they are, and b) how to defactor them. The answer to a) is that they are just the minor determinants (or "minors" for short) of rectangular matrices, or in other words, the closest thing rectangular matrices such as mappings have to a real determinant. And the answer to b) is that you simply extract the GCD of the entries in this list of minors. So if defactoring a list of minor determinants means dividing common factors out, then it should be little surprise that achieving a real determinant of ±1 is equivalent to defactoring, and thereby that leveraging the unimodularity of the other matrix produced by the Hermite decomposition should be valuable in this capacity.