{
  "id": "math/0404349",
  "version": "v1",
  "published": "2004-04-19T22:16:19.000Z",
  "updated": "2004-04-19T22:16:19.000Z",
  "title": "On the Higher-Order Derivatives of Spectral Functions: Two Special Cases",
  "authors": [
    "Hristo S. Sendov"
  ],
  "comment": "27 pages",
  "categories": [
    "math.OC",
    "math.SP"
  ],
  "abstract": "In this work we use the tensorial language developed in [8] and [9] to differentiate functions of eigenvalues of symmetric matrices. We describe the formulae for the k-th derivative of such functions in two cases. The first case concerns the derivatives of the composition of an arbitrary differentiable function with the eigenvalues at a matrix with distinct eigenvalues. The second development describes the derivatives of the composition of a separable symmetric function with the eigenvalues at an arbitrary symmetric matrix. In the concluding section we re-derive the formula for the Hessian of a general spectral function at an arbitrary point. Our approach leads to a shorter, streamlined derivation than the original in [6]. The language we use, based on the generalized Hadamard product, allows us to view the differentiation of spectral functions as a routine calculus-type procedure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-04-19T22:16:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49R50",
      "47A75",
      "15A18",
      "15A69"
    ],
    "keywords": [
      "special cases",
      "higher-order derivatives",
      "routine calculus-type procedure",
      "arbitrary symmetric matrix",
      "general spectral function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4349S"
    }
  }
}