{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-calc-matrix",
    "area": "mathematics",
    "topic": "calculus",
    "origin": "original",
    "title": "Matrix gradient and Hessian",
    "skills": [
      "matrix-calculus",
      "gradient",
      "hessian",
      "convexity"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 10,
    "prerequisites": [
      "matrix-multiplication",
      "chain-rule"
    ],
    "prompt": "For\n$$\nf(w) = \\lVert Xw-y \\rVert_2^2 + \\lambda \\lVert w \\rVert_2^2,\n$$\nderive the gradient and Hessian. State a sufficient condition for strict convexity.\n",
    "answer": "$$\n\\nabla f(w) = 2X^{\\mathsf T}(Xw-y) + 2\\lambda w,\n\\qquad\n\\nabla^2 f(w) = 2X^{\\mathsf T}X + 2\\lambda I.\n$$\nIt is strictly convex if $\\lambda>0$, or if $X$ has full column rank when $\\lambda=0$.\n",
    "check": {
      "kind": "symbolic",
      "id": "original-calc-matrix",
      "values": [
        2,
        2
      ]
    }
  }
}