{
  "id": "1805.04488",
  "version": "v1",
  "published": "2018-05-11T16:50:04.000Z",
  "updated": "2018-05-11T16:50:04.000Z",
  "title": "Standard Triples for Algebraic Linearizations of Matrix Polynomials",
  "authors": [
    "Eunice Y. S. Chan",
    "Robert M. Corless",
    "Leili Rafiee Sevyeri"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NA"
  ],
  "abstract": "Standard triples $\\mathbf{X}$, $\\mathbf{C}_{1} - \\mathbf{C}_{0}$, $\\mathbf{Y}$ of nonsingular matrix polynomials $\\mathbf{P}(z) \\in \\mathbb{C}^{r \\times r}$ have the property $\\mathbf{X}(z \\mathbf{C}_{1}~-~\\mathbf{C}_{0})^{-1}\\mathbf{Y}~=~\\mathbf{P}^{-1}(z)$ for $z \\notin \\Lambda(\\mathbf{P}(z))$. They can be used in constructing algebraic linearizations; for example, for $\\mathbf{h}(z) = z \\mathbf{a}(z)\\mathbf{b}(z) + c \\in \\mathbb{C}^{r \\times r}$ from linearizations for $\\mathbf{a}(z)$ and $\\mathbf{b}(z)$. We tabulate standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of a transposed linearization, a flipped linearization, and a transposed-and-flipped linearization. We give proofs for the less familiar bases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-11T16:50:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65F15",
      "15A22",
      "65D05"
    ],
    "keywords": [
      "orthogonal polynomial bases",
      "nonsingular matrix polynomials",
      "hermite interpolational bases",
      "newton interpolational bases",
      "tabulate standard triples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}