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

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And you've got your multimap!

Another way to think about this is that you've found all of the '''[https://en.wikipedia.org/wiki/Minor_(linear_algebra) minors]''' of the matrix, where a minor is the determinant of a square submatrix. So we can think of multimaps as lists of minors.

Another way to think about this is that you've found all of the '''[https://en.wikipedia.org/wiki/Minor_(linear_algebra) minors]''' of the matrix, where a minor is the determinant of a square submatrix. So we can think of multimaps as lists of minors<ref>This operation is called the wedge product. Another way to think of it is as the difference between two outer products. For example, consider wedging {{map|7 11 16}} with {{map|12 19 28}}. Call these u and v, respectively. So u∧v (that's the wedge product) is the same thing as u.v - v.u (where the dots are outer products). Those two outer products are:

{{vector|{{covector|84 133 196}} {{covector|132 209 308}} {{covector|192 304 448}}}}

{{vector|{{covector|84 132 192}} {{covector|133 209 304}} {{covector|196 308 448}}}}

And so it's clear to see that the difference is

{{vector|{{covector|0 1 4}} {{covector|-1 0 4}} {{covector|-4 -4 0}}}}

So it's really just a different way of slicing and dicing a bunch of determinants. With this difference of outer products approach, you do all the multiplications at once as the first step, then all of the differences at once as the second step. Whereas with the minors approach, you do a ton of separate differences of products individually.</ref>.

Now, if you want the canonical multimap, two further steps are required:

@@ Line 1,188: / Line 1,188: @@
 And you've got your multimap!
-Another way to think about this is that you've found all of the '''[https://en.wikipedia.org/wiki/Minor_(linear_algebra) minors]''' of the matrix, where a minor is the determinant of a square submatrix. So we can think of multimaps as lists of minors.
+Another way to think about this is that you've found all of the '''[https://en.wikipedia.org/wiki/Minor_(linear_algebra) minors]''' of the matrix, where a minor is the determinant of a square submatrix. So we can think of multimaps as lists of minors<ref>This operation is called the wedge product. Another way to think of it is as the difference between two outer products. For example, consider wedging {{map|7 11 16}} with {{map|12 19 28}}. Call these u and v, respectively. So u∧v (that's the wedge product) is the same thing as u.v - v.u (where the dots are outer products). Those two outer products are:<br>
+<br>
+{{vector|{{covector|84 133 196}} {{covector|132 209 308}} {{covector|192 304 448}}}}<br>
+{{vector|{{covector|84 132 192}} {{covector|133 209 304}} {{covector|196 308 448}}}}<br>
+<br>
+And so it's clear to see that the difference is<br>
+<br>
+{{vector|{{covector|0 1 4}} {{covector|-1 0 4}} {{covector|-4 -4 0}}}}<br>
+<br>
+So it's really just a different way of slicing and dicing a bunch of determinants. With this difference of outer products approach, you do all the multiplications at once as the first step, then all of the differences at once as the second step. Whereas with the minors approach, you do a ton of separate differences of products individually.</ref>.
 Now, if you want the canonical multimap, two further steps are required: