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

Line 1,569:

|(column) vector, matrix column

|-

!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of bra-ket notation.</ref>

!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of bra-ket notation. In other words, you can think of the multi(co)vectors used in RTT as compressed forms of antisymmetric tensors. A tensor is like a multi-dimensional array, or you could think of it as a generalization of matrices to other dimensions besides 2. And confusingly, antisymmetric doesn't mean the non-symmetric or undoing symmetry; it is a specific type of symmetry. This is made clearer in another name for antisymmetric, which is skew-symmetric. This reveals that the type of symmetry these tensors exhibit is along a skew, or diagonal. Let's take the example of meantone. Technically speaking, when you take the wedge product of two vectors (or covectors), the result is a matrix:

<math>

\left[ \begin{array} {rrr}

0 & 1 & 4 \\

-1 & 0 & 4 \\

-4 & -4 & 0 \\

\end{array} \right]

</math>

But this isn't a particularly efficient representation of this information. Everything in the bottom left triangular half is a mirror of what's in the top right triangular half, just with the signs changed! We compress this as {{multicovector|1 4 4}} instead, leveraging the antisymmetry. For a higher dimensional temperament, the tensor representation would be a higher-dimensional square, such as a cube, hypercube, 5-cube, etc. and the antisymmetry would lead to one tetrahedral, 4-simplex, 5-simplex, etc. half being mirrored but with signs changed. So we compress it into a multicovector with even more brackets. No big deal.

</ref>

|covector

|list of covectors

@@ Line 1,569: / Line 1,569: @@
 |(column) vector, matrix column
 |-
-!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of bra-ket notation.</ref>
+!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of bra-ket notation. In other words, you can think of the multi(co)vectors used in RTT as compressed forms of antisymmetric tensors. A tensor is like a multi-dimensional array, or you could think of it as a generalization of matrices to other dimensions besides 2. And confusingly, antisymmetric doesn't mean the non-symmetric or undoing symmetry; it is a specific type of symmetry. This is made clearer in another name for antisymmetric, which is skew-symmetric. This reveals that the type of symmetry these tensors exhibit is along a skew, or diagonal. Let's take the example of meantone. Technically speaking, when you take the wedge product of two vectors (or covectors), the result is a matrix:
+<math>
+\left[ \begin{array} {rrr}
+& 1 & 4 \\
+-1 & 0 & 4 \\
+-4 & -4 & 0 \\
+\end{array} \right]
+</math>
+But this isn't a particularly efficient representation of this information. Everything in the bottom left triangular half is a mirror of what's in the top right triangular half, just with the signs changed! We compress this as {{multicovector|1 4 4}} instead, leveraging the antisymmetry. For a higher dimensional temperament, the tensor representation would be a higher-dimensional square, such as a cube, hypercube, 5-cube, etc. and the antisymmetry would lead to one tetrahedral, 4-simplex, 5-simplex, etc. half being mirrored but with signs changed. So we compress it into a multicovector with even more brackets. No big deal.
+</ref>
 |covector
 |list of covectors