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    "---\n",
    "title: Biostat 216 Homework 6\n",
    "subtitle: 'Due Nov 26 @ 11:59pm'\n",
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   "source": [
    "Submit a PDF (scanned/photographed from handwritten solutions, or converted from RMarkdown or Jupyter Notebook) to Gracescope on BruinLearn."
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   "source": [
    "## Linear equations, matrix inverses, orthogonal projection\n",
    "\n",
    "### BV exercises\n",
    "\n",
    "BV 11.5, 11.12, 11.18"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Q1 \n",
    "\n",
    "Find the orthogonal projector into the plane spanned by the vectors $\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix}$ and $\\begin{pmatrix} -2 \\\\ 2 \\\\ 1 \\end{pmatrix}$.   \n",
    "\n",
    "Find the orthogonal projection of the point $\\mathbf{1}_3$ into the plane spanned by the vectors $\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix}$ and $\\begin{pmatrix} -2 \\\\ 2 \\\\ 1 \\end{pmatrix}$. "
   ]
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   "metadata": {},
   "source": [
    "### Q2\n",
    "\n",
    "Matrices that satisfy $\\mathbf{A}' \\mathbf{A} = \\mathbf{A} \\mathbf{A}'$ are said to be _normal_. Give an example of asymmetric (not symmetric), normal matrix. If $\\mathbf{A}$ is normal, then prove that every vector in $\\mathcal{C}(\\mathbf{A})$ is orthogonal to every vector in $\\mathcal{N}(\\mathbf{A})$. "
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   "metadata": {},
   "source": [
    "### Q3\n",
    "\n",
    "Prove the following facts about triangular and orthogonal matrices.\n",
    "\n",
    "1. The product of two upper (lower) triangular matrices is upper (lower) triangular.\n",
    "    \n",
    "2. The inverse of an upper (lower) triangular matrix is upper (lower) triangular.\n",
    "    \n",
    "3. The product of two unit upper (lower) triangular matrices is unit upper (lower) triangular.\n",
    "    \n",
    "4. The inverse of a unit upper (lower) triangular matrix is unit upper (lower) triangular.\n",
    "    \n",
    "5. An orthogonal upper (lower) triangular matrix is diagonal."
   ]
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   "source": [
    "## Determinant\n",
    "\n",
    "### Q4\n",
    "\n",
    "Determinant.\n",
    "\n",
    "1. Find the determinant of the following two matrices without doing any computations:\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "0 & 0 & 1\\\\\n",
    "0 & 1 & 0 \\\\\n",
    "1 & 0 & 0\n",
    "\\end{pmatrix}, \\quad \\begin{pmatrix}\n",
    "0 & 0 & 1 & 0 \\\\\n",
    "0 & 1 & 0 & 0 \\\\\n",
    "1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1\n",
    "\\end{pmatrix}.\n",
    "$$  \n",
    "\n",
    "2. Let $\\mathbf{A} \\in \\mathbb{R}^{5 \\times 5}$ with $\\det(\\mathbf{A}) = -3$. Find $\\det(\\mathbf{A}^3)$, $\\det(\\mathbf{A}^{-1})$, and $\\det(2\\mathbf{A})$.  \n",
    "    \n",
    "3. Find the determinant of the matrix\n",
    "$\\begin{pmatrix}\n",
    "0 & 0 & 1 \\\\\n",
    "2 & 3 & 4 \\\\\n",
    "0 & 5 & 6\n",
    "\\end{pmatrix}$."
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   "source": [
    "## Eigenvalues and eigenvectors\n",
    "\n",
    "### Q5\n",
    "\n",
    "Diagonalize (show the steps to find eigenvalues and eigenvectors)\n",
    "$$\n",
    "\\mathbf{A} = \\begin{pmatrix} 2 & -1 \\\\ -1 & 2 \\end{pmatrix}\n",
    "$$\n",
    "and compute $\\mathbf{X} \\boldsymbol{\\Lambda}^k \\mathbf{X}^{-1}$ to prove the formula\n",
    "$$\n",
    "\\mathbf{A}^k = \\frac 12 \\begin{pmatrix} 1 + 3^k & 1 - 3^k \\\\ 1 - 3^k & 1 + 3^k \\end{pmatrix}.\n",
    "$$"
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   "source": [
    "### Q6\n",
    "\n",
    "Suppose the eigenvalues of a square matrix $\\mathbf{A} \\in \\mathbb{R}^{n \\times n}$ are $\\lambda_1, \\ldots, \\lambda_n$. Show that $\\det (\\mathbf{A}) = \\prod_{i=1}^n \\lambda_i$. Hint: $\\lambda_i$s are roots of the characteristic polynomial."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Q7\n",
    "\n",
    "Ture of false. For each statement, indicate it is true or false and gives a brief explanation.\n",
    "\n",
    "1. If the columns of $\\mathbf{X}$ (eigenvectors of a square matrix $\\mathbf{A}$) are linearly independent, then (a) $\\mathbf{A}$ is invertible; (b) $\\mathbf{A}$ is diagonalizable; (c) $\\mathbf{X}$ is invertible; (d) $\\mathbf{X}$ is diagonalizable.\n",
    "    \n",
    "2. If the eigenvalues of $\\mathbf{A}$ are 2, 2, 5 then the matrix is certainly (a) invertible; (b) diagonalizable. \n",
    "    \n",
    "3. If the only eigenvectors of $\\mathbf{A}$ are multiples of $\\begin{pmatrix} 1 \\\\ 4 \\end{pmatrix}$, then the matrix has (a) no inverse; (b) a repeated eigenvalue; (c) no diagonalization $\\mathbf{X} \\boldsymbol{\\Lambda} \\mathbf{X}^{-1}$. "
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   "metadata": {},
   "source": [
    "### Q8\n",
    "\n",
    "Let $\\mathbf{A} \\in \\mathbb{R}^{m \\times n}$ and $\\mathbf{B} \\in \\mathbb{R}^{n \\times m}$. Show that $\\mathbf{A} \\mathbf{B}$ and $\\mathbf{B} \\mathbf{A}$ share the same non-zero eigenvalues. Hint: examine the eigen-equations."
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   "source": [
    "### Q9\n",
    "\n",
    "Let\n",
    "$$\n",
    "\\mathbf{A} = \\begin{pmatrix}\n",
    "0 & 2 \\\\\n",
    "1 & 1\n",
    "\\end{pmatrix}, \\quad \\mathbf{B} = \\begin{pmatrix}\n",
    "1 & 2 \\\\\n",
    "0 & 1\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "1. Find eigenvalues and eigenvectors of $\\mathbf{A}$ and $\\mathbf{A}^{-1}$. Do they have same eigenvectors? What's the relationship between their eigenvalues?   \n",
    "\n",
    "2. Find eigenvalues of $\\mathbf{B}$ and $\\mathbf{A} + \\mathbf{B}$. Are eigenvalues of $\\mathbf{A} + \\mathbf{B}$ equal to eigenvalues of $\\mathbf{A}$ plus eigenvalues of $\\mathbf{B}$?   \n",
    "\n",
    "3. Find eigenvalues of $\\mathbf{A} \\mathbf{B}$ and $\\mathbf{B} \\mathbf{A}$. Are the eigenvalues of $\\mathbf{A} \\mathbf{B}$ equal to eigenvalues of $\\mathbf{A}$ times eigenvalues of $\\mathbf{B}$? Are the eigenvalues of $\\mathbf{A} \\mathbf{B}$ equal to eigenvalues of $\\mathbf{B} \\mathbf{A}$?"
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   "source": [
    "### Q10\n",
    "\n",
    "Suppose $\\mathbf{A}$ has eigenvalues 0, 3, 5 with independent eigenvectors $\\mathbf{u}$, $\\mathbf{v}$, $\\mathbf{w}$ respectively. \n",
    "\n",
    "1. Give a basis for $\\mathcal{C}(\\mathbf{A})$ and a basis for $\\mathcal{N}(\\mathbf{A})$.\n",
    "\n",
    "2. Find a particular solution to $\\mathbf{A} \\mathbf{x} = \\mathbf{v} + \\mathbf{w}$. Find all solutions.  \n",
    "\n",
    "3. Show that the linear equation $\\mathbf{A} \\mathbf{x} = \\mathbf{u}$ has no solution. "
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