When the Tracy-Singh product of matrices represents a certain operation on linear operators
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909395123699712 |
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| author | Chouraqui, Fabienne |
| author_facet | Chouraqui, Fabienne |
| contents | Given two linear transformations, with representing matrices $A$ and $B$ with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices $A$ and $B$ corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that each matrix is a square matrix of order of the form $n^2$, $n>1$, and is partitioned into $n^2$ square blocks of order $n$, then their Tracy-Singh product, $A \boxtimes B$, is similar to $A \otimes B$, and the change of basis matrix is a permutation matrix. In this note, we prove that in the special case of linear operators induced from set-theoretic solutions of the Yang-Baxter equation, the Tracy-Singh product of their representing matrices is the representing matrix of the linear operator obtained from the direct product of the set-theoretic solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01342 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | When the Tracy-Singh product of matrices represents a certain operation on linear operators Chouraqui, Fabienne Combinatorics Rings and Algebras Given two linear transformations, with representing matrices $A$ and $B$ with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices $A$ and $B$ corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that each matrix is a square matrix of order of the form $n^2$, $n>1$, and is partitioned into $n^2$ square blocks of order $n$, then their Tracy-Singh product, $A \boxtimes B$, is similar to $A \otimes B$, and the change of basis matrix is a permutation matrix. In this note, we prove that in the special case of linear operators induced from set-theoretic solutions of the Yang-Baxter equation, the Tracy-Singh product of their representing matrices is the representing matrix of the linear operator obtained from the direct product of the set-theoretic solutions. |
| title | When the Tracy-Singh product of matrices represents a certain operation on linear operators |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2305.01342 |