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    "---\n",
    "title: Biostat 216 Homework 2\n",
    "subtitle: 'Due Oct 11 @ 11:59pm'\n",
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    "Submit a PDF (scanned/photographed from handwritten solutions, or converted from RMarkdown or Jupyter Notebook or Quarto) to Gradescope in BruinLearn."
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    "## Q1. Sub-multiplicity of Frobenius norm\n",
    "\n",
    "Show the matrix norm property \n",
    "$$\n",
    "\\|\\mathbf{A} \\mathbf{B}\\|_{\\text{F}} \\le \\|\\mathbf{A}\\|_{\\text{F}} \\|\\mathbf{B}\\|_{\\text{F}}\n",
    "$$ \n",
    "for the Frobenius norm. Hint: Cauchy-Schwartz inequality.\n",
    "\n",
    "BV exercise 6.14 is a special case of this result."
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   "source": [
    "## Q2. Induced matrix norm\n",
    "\n",
    "For any vector norm $\\|\\mathbf{x}\\|$ on $\\mathbb{R}^m$ and $\\mathbb{R}^n$, there is an induced matrix norm $\\|\\mathbf{A}\\|$ on $m \\times n$ matrices defined by\n",
    "$$\n",
    "\\|\\mathbf{A}\\| = \\sup_{\\mathbf{x} \\ne \\mathbf{0}} \\frac{\\|\\mathbf{A} \\mathbf{x}\\|}{\\|\\mathbf{x}\\|} = \\sup_{\\|\\mathbf{x}\\|=1} \\|\\mathbf{A} \\mathbf{x}\\|.\n",
    "$$\n",
    "\n",
    "1. Show the second equality in the above equation.\n",
    "\n",
    "2. Show the four properties (positive definiteness, homogeneity, triangle inequality, sub-multiplicity) for the induced matrix norm.\n",
    "\n",
    "3. Show that the Frobenius norm is different from the induced matrix-2 norm."
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   "source": [
    "## BV exercises\n",
    "\n",
    "5.1, 5.2, 5.4, 5.6, 5.8, 5.9, 6.3, 6.11, 6.17, 6.21, 6.22"
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