definition of same-length multi-dimensional array antisymmetrized with respect to set of indexes
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 0: Note 1
- 1: Structured Description
- 2: Natural Language Description
- 3: Note 2
Starting Context
Target Context
- The reader will have a definition of same-length multi-dimensional array antisymmetrized with respect to set of indexes.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
0: Note 1
Any \(n\)-length 2-dimensional array is an \(n \times n\) matrix.
The components of any \((p-q)\) tensor with respect to any bases is a vectors-space-dimension-length (p + q)-dimensional array, but a same-length multi-dimensional array is not necessarily the components of a tensor: a same-length multi-dimensional array is in general just a collection of numbers not necessarily related to any tensor.
The dimensions of the same-length multi-dimensional array have to have the same length for our purpose, because otherwise, a permutation of indexes would not make sense.
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M\): \(\in \{\text{ the same-length } n \text{ -dimensional arrays } \}\), \(= \begin{pmatrix} M_{j_1, ..., j_n} \end{pmatrix}\)
\( S\): \(\subseteq \{1, ..., n\}\), \(= \{l_1, ..., l_k\}\)
\(*M'\): \(\in \{\text{ the same-length } n \text{ -dimensional arrays }\}\), \(= \begin{pmatrix} M'_{j_1, ..., j_n} \end{pmatrix} = \begin{pmatrix} 1 / k! \sum_\sigma sgn \sigma M_{\sigma ((j_1, ..., j_n))_1, ..., \sigma ((j_1, ..., j_n))_n} \end{pmatrix}\), where each \(\sigma\) is a permutation of \((j_1, ..., j_n)\) such that only \((j_{l_1}, ..., j_{l_k})\) is permutated
//
Conditions:
//
2: Natural Language Description
For any same-length \(n\)-dimensional array, \(M = \begin{pmatrix} M_{j_1, ..., j_n} \end{pmatrix}\), and any subset, \(S = \{l_1, ..., l_k\} \subseteq \{1, ..., n\}\), the array, \(M' = \begin{pmatrix} M'_{j_1, ..., j_n} \end{pmatrix} = \begin{pmatrix} 1 / k! \sum_\sigma sgn \sigma M_{\sigma ((j_1, ..., j_n))_1, ..., \sigma ((j_1, ..., j_n))_n} \end{pmatrix}\), where each \(\sigma\) is a permutation of \((j_1, ..., j_n)\) such that only \((j_{l_1}, ..., j_{l_k})\) is permutated
3: Note 2
When, for example, \(S = \{1, ..., k\}\), the antisymmetrized array is denoted as \(\begin{pmatrix} M_{[j_1, ..., j_k], j_{k + 1}, ..., j_n} \end{pmatrix}\).
For any permutation, \(\sigma'\), that permutates only \((j_{l_1}, ..., j_{l_k})\), \(M'_{\sigma' ((j_1, ..., j_n))_1, ..., \sigma' ((j_1, ..., j_n))_n} = sgn \sigma' M'_{j_1, ..., j_n}\), which is indeed the purpose of antisymmetrizing, because \(M'_{\sigma' ((j_1, ..., j_n))_1, ..., \sigma' ((j_1, ..., j_n))_n} = 1 / k! \sum_\sigma sgn \sigma M_{\sigma \circ \sigma' ((j_1, ..., j_n))_1, ..., \sigma \circ \sigma' ((j_1, ..., j_n))_n} = 1 / k! \sum_\sigma sgn \sigma sgn \sigma' sgn \sigma' M_{\sigma \circ \sigma' ((j_1, ..., j_n))_1, ..., \sigma \circ \sigma' ((j_1, ..., j_n))_n} = sgn \sigma' 1 / k! \sum_\sigma sgn (\sigma \circ \sigma') M_{\sigma \circ \sigma' ((j_1, ..., j_n))_1, ..., \sigma \circ \sigma' ((j_1, ..., j_n))_n} = sgn \sigma' 1 / k! \sum_{\sigma \circ \sigma'} sgn (\sigma \circ \sigma') M_{\sigma \circ \sigma' ((j_1, ..., j_n))_1, ..., \sigma \circ \sigma' ((j_1, ..., j_n))_n}\), because as \(\sigma\) iterates all the permutations of \(S\), also \(\sigma \circ \sigma'\) iterates all the permutations of \(S\), by the proposition that any permutation bijectively maps the set of all the permutations onto the set of all the permutations by composition from left or right, \(= sgn \sigma' 1 / k! \sum_{\sigma''} sgn (\sigma'') M_{\sigma'' ((j_1, ..., j_n))_1, ..., \sigma'' ((j_1, ..., j_n))_n}\), with \(\sigma \circ \sigma'\) just renamed as \(\sigma''\), \(= sgn \sigma' M'_{j_1, ..., j_n}\).
