{
  "id": "2305.01342",
  "version": "v1",
  "published": "2023-05-02T11:42:13.000Z",
  "updated": "2023-05-02T11:42:13.000Z",
  "title": "When the Tracy-Singh product of matrices represents a certain operation on linear operators",
  "authors": [
    "Fabienne Chouraqui"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2303.02964",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-02T11:42:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tracy-singh product",
      "linear operator",
      "matrices represents",
      "representing matrix",
      "set-theoretic solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}