Defactoring: Difference between revisions

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One reason for doing a column Hermite decomposition on a mapping can be understood by comparing the sizes of the unimodular matrices. Matrices are often described as <math>m×n</math>, where <math>m</math> is the row count and <math>n</math> is the column count. In the case of mappings it may be superior to use variable names corresponding to the domain concepts of rank <math>r</math>, and dimension <math>d</math>, i.e. to speak of <math>r×d</math> mappings. The key bit of info here is that — for non-trivial mappings, anyway — <math>d</math> is always greater than <math>r</math>. So a standard row-based Hermite decomposition, i.e. to the right, is going to produce an <math>r×r</math> unimodular matrix, while a column-based Hermite decomposition, i.e. to the bottom, is going to produce a <math>d×d</math> unimodular matrix. For example, 5-limit meantone has <math>r=2</math> and <math>d=3</math>, so a standard row-based Hermite decomposition is going to produce a unimodular matrix that is only 2×2, while the column-based Hermite decomposition will produce one that is 3×3. With <math>d>r</math>, it's clear that the column-based decomposition in general will always produced the larger unimodular matrix. In fact, the row-based decomposition is too small to be capable of enclosing an amount of entries equal to the count of entries in the original mapping, and therefore it could never support preserving the entirety of the important information from the input (in terms of our example, a 3×3 matrix can hold a 2×3 matrix, but a 2×2 matrix cannot).

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.

==== relationship with Smith defactoring ====

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 One reason for doing a column Hermite decomposition on a mapping can be understood by comparing the sizes of the unimodular matrices. Matrices are often described as <span><math>m×n</math></span>, where <span><math>m</math></span> is the row count and <span><math>n</math></span> is the column count. In the case of mappings it may be superior to use variable names corresponding to the domain concepts of rank <span><math>r</math></span>, and dimension <span><math>d</math></span>, i.e. to speak of <span><math>r×d</math></span> mappings. The key bit of info here is that — for non-trivial mappings, anyway — <span><math>d</math></span> is always greater than <span><math>r</math></span>. So a standard row-based Hermite decomposition, i.e. to the right, is going to produce an <span><math>r×r</math></span> unimodular matrix, while a column-based Hermite decomposition, i.e. to the bottom, is going to produce a <span><math>d×d</math></span> unimodular matrix. For example, 5-limit meantone has <span><math>r=2</math></span> and <span><math>d=3</math></span>, so a standard row-based Hermite decomposition is going to produce a unimodular matrix that is only 2×2, while the column-based Hermite decomposition will produce one that is 3×3. With <span><math>d>r</math></span>, it's clear that the column-based decomposition in general will always produced the larger unimodular matrix. In fact, the row-based decomposition is too small to be capable of enclosing an amount of entries equal to the count of entries in the original mapping, and therefore it could never support preserving the entirety of the important information from the input (in terms of our example, a 3×3 matrix can hold a 2×3 matrix, but a 2×2 matrix cannot).
+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.
 ==== relationship with Smith defactoring ====