Biostat 216 Homework 5

Due Nov 4 Friday @ 11:59pm

Submit a PDF (scanned/photographed from handwritten solutions, or converted from RMarkdown or Jupyter Notebook) to Gracescope on BruinLearn.

BV 11.3, 11.4, 11.5, 11.9, 11.12, 11.28

  • Q1. Let $\mathbf{A} \in \mathbb{R}^{n \times n}$ be a symmetric matrix. Prove that $\langle \mathbf{A} \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{x}, \mathbf{A} \mathbf{y} \rangle$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$. Give an example that it is not necessary true if $\mathbf{A}$ is not symmetric.

  • Q2. Find the orthogonal projector into the plane spanned by the vectors $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ 2 \\ 1 \end{pmatrix}$.

    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 $\begin{pmatrix} -2 \\ 2 \\ 1 \end{pmatrix}$.

  • Q3. 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 $\mathcal{N}(\mathbf{A})$.
  • Q4. Let $\mathbf{A}$ be a symmetric matrix. Show that the system $\mathbf{A} \mathbf{x} = \mathbf{b}$ has a solution if and only if $\mathbf{b}$ is orthogonal to $\mathcal{N}(\mathbf{A})$.
  • Q5. Determinant.
    • (1). Find the determinant of the following two matrices without doing any computations: $$ \begin{pmatrix} 0 & 0 & 1\\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$
    • (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 $\det(2\mathbf{A})$.
    • (3). Find the determinant of the matrix $\begin{pmatrix} 0 & 0 & 1 \\ 2 & 3 & 4 \\ 0 & 5 & 6 \end{pmatrix}$. Hint: find the row and column permutations that make $\mathbf{A}$ triangular; then use product rule.
  • Q6. Let $$ \mathbf{A} = \begin{pmatrix} 0 & 2 \\ 1 & 1 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} $$
    • (1). Find eigenvalues and eigenvectors of $\mathbf{A}$ and $\mathbf{A}^{-1}$. Do they have same eigenvectors? What's the relationship between their eigenvalues?
    • (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 $\mathbf{B}$?
    • (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 $\mathbf{B} \mathbf{A}$?