Defactoring: Difference between revisions

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After we know how to do these two things individually, we'll learn how to tweak them and assemble them together in order to perform a complete column Hermite defactoring.

Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[~~User:Cmloegcmluin/RTT_How~~-To#null-space|here]]):

Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[Douglas_Blumeyer%27s_RTT_How-To#null-space|here]]):

# augmenting your matrix with an identity matrix

# performing elementary row or column operations until a desired state is achieved<ref>For convenience, here is a summary of the three different techniques and their targets:<br>

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= canonical comma-bases =

Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[~~User:Cmloegcmluin/RTT How~~-To#null-space]]</ref>once at the beginning, and then antitransposed again at the end.

Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#null-space]]</ref>once at the beginning, and then antitransposed again at the end.

For example, suppose we have the comma-basis for septimal meantone:

@@ Line 319: / Line 319: @@
 After we know how to do these two things individually, we'll learn how to tweak them and assemble them together in order to perform a complete column Hermite defactoring.
-Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[User:Cmloegcmluin/RTT_How-To#null-space|here]]):
+Fortunately, both of these two processes can be done using a technique you may already be familiar with if you've learned how to calculate the null-space of a mapping by hand (as demonstrated [[Douglas_Blumeyer%27s_RTT_How-To#null-space|here]]):
 # augmenting your matrix with an identity matrix
 # performing elementary row or column operations until a desired state is achieved<ref>For convenience, here is a summary of the three different techniques and their targets:<br>
@@ Line 863: / Line 863: @@
 = canonical comma-bases =
-Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[User:Cmloegcmluin/RTT How-To#null-space]]</ref>once at the beginning, and then antitransposed again at the end.
+Canonical form is not only for mappings; comma-bases may also be put into canonical form. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#null-space]]</ref>once at the beginning, and then antitransposed again at the end.
 For example, suppose we have the comma-basis for septimal meantone: