Normal forms: Difference between revisions
More accurate describe the "shift" form since the algorithm is working as intended now |
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The '''Hermite normal form''', or HNF for short, is an important normal form that is defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory. | The '''Hermite normal form''', or HNF for short, is an important normal form that is defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory. | ||
The Hermite normal form is unique. Besides that, it is also the integer analogue of the reduced | The Hermite normal form is unique. Besides that, it is also the integer analogue of the reduced echelon form, so it can be used to solve systems of equations in the integers by Gaussian elimination. | ||
There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows. An ''n''×''m'' integer matrix ''H'' is in (row-wise) HNF if it satisfies the following conditions: | There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows. An ''n''×''m'' integer matrix ''H'' is in (row-wise) HNF if it satisfies the following conditions: | ||