Douglas Blumeyer's RTT How-To: Difference between revisions

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If you take the ~~Hermite~~ canonical form of {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, that’s what you get. It’s also what you get if you take the ~~Hermite~~ canonical form of {{vector|{{map|12 19 28}} {{map|19 30 44}}}}, etc. That’s the power of canonicalization.

If you take the integer canonical form of {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, that’s what you get. It’s also what you get if you take the integer canonical form of {{vector|{{map|12 19 28}} {{map|19 30 44}}}}, etc. That’s the power of canonicalization.

To find the ~~Hermite~~ canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and anti-null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.

To find the integer canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and anti-null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.

Unfortunately, Wolfram Alpha is incapable of computing this in one go. For that, you'll need up your game to Wolfram computable notebooks. It only takes a moment and you can sign up for free.

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-If you take the Hermite canonical form of {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, that’s what you get. It’s also what you get if you take the Hermite canonical form of {{vector|{{map|12 19 28}} {{map|19 30 44}}}}, etc. That’s the power of canonicalization.
+If you take the integer canonical form of {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, that’s what you get. It’s also what you get if you take the integer canonical form of {{vector|{{map|12 19 28}} {{map|19 30 44}}}}, etc. That’s the power of canonicalization.
-To find the Hermite canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and anti-null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.
+To find the integer canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and anti-null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.
 Unfortunately, Wolfram Alpha is incapable of computing this in one go. For that, you'll need up your game to Wolfram computable notebooks. It only takes a moment and you can sign up for free.