Matrix echelon forms: Difference between revisions
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==IRREF== | ==IRREF== | ||
'''[[Normal_lists|Integer Reduced Row Echelon Form]]''', or '''IRREF''': based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an [https://en.wikipedia.org/wiki/Euler_diagram Euler diagram] (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That's because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers. | '''[[Normal_lists|Integer Reduced Row Echelon Form]]''', or '''IRREF''': based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an [https://en.wikipedia.org/wiki/Euler_diagram Euler diagram] (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That's because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers. Of course, this sometimes results in the pivots no longer being 1, so sometimes it is no longer RREF. It is always still REF, though,<ref>Also, it will always still satisfy the second aspect of reduced, i.e. that all other entries in pivot columns besides the pivots are zeroes.</ref> and because it is also always integer, that makes it always IREF; therefore, IRREF is strictly a subcategory of IREF. And because the RREF is unique, and the conversion process does not alter that, the IRREF is also unique. | ||
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There is also the '''[https://en.wikipedia.org/wiki/Smith_normal_form Smith Normal Form]''', or '''SNF''', but we won't be discussing it in this context, because putting a mapping into SNF obliterates a lot of meaningful RTT information. SNF is also echelon, and integer, so like HNF it is also always IREF. But SNF requires that every single entry other than the pivots are zero, and that the pivots all fall exactly along the main diagonal of the matrix. The SNF essentially reduces a matrix down to the information of what its rank is and whether it is enfactored. For example, all 5-limit rank-2 temperaments such as meantone, porcupine, mavila, hanson, etc. have the same SNF: {{rket|{{map|1 0 0}} {{map|0 1 0}}}}. Or, if you 2-enfactor them, they will all have the SNF {{rket|{{map|1 0 0}} {{map|0 2 0}}}}. So while the SNF is closely related to defactoring, it is not itself a useful form to put mappings into.<ref>Here is a useful resource for computing the Smith Normal Form manually, if you are interested: https://math.stackexchange.com/questions/133076/computing-the-smith-normal-form The fact that it involves calculating so many GCDs is unsurprising given its ability to defactor matrices.</ref> | There is also the '''[https://en.wikipedia.org/wiki/Smith_normal_form Smith Normal Form]''', or '''SNF''', but we won't be discussing it in this context, because putting a mapping into SNF obliterates a lot of meaningful RTT information. SNF is also echelon, and integer, so like HNF it is also always IREF. But SNF requires that every single entry other than the pivots are zero, and that the pivots all fall exactly along the main diagonal of the matrix. The SNF essentially reduces a matrix down to the information of what its rank is and whether it is enfactored. For example, all 5-limit rank-2 temperaments such as meantone, porcupine, mavila, hanson, etc. have the same SNF: {{rket|{{map|1 0 0}} {{map|0 1 0}}}}. Or, if you 2-enfactor them, they will all have the SNF {{rket|{{map|1 0 0}} {{map|0 2 0}}}}. So while the SNF is closely related to defactoring, it is not itself a useful form to put mappings into.<ref>Here is a useful resource for computing the Smith Normal Form manually, if you are interested: https://math.stackexchange.com/questions/133076/computing-the-smith-normal-form The fact that it involves calculating so many GCDs is unsurprising given its ability to defactor matrices.</ref> | ||
The SNF is used for finding [[ | The SNF is used for finding [[generator preimage transversal]]s. | ||
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