Math 487, Homework 1

$latex \def\Real{\mathbb{R}}\def\ee{\mathbf{e}}\def\ff{\mathbf{f}}\newcommand\ket[1]{\vert{#1}\rangle}\newcommand\bra[1]{\langle{#1}\vert}\newcommand\bk[2]{\langle #1\vert#2\rangle}\def\lin{\mathtt{Lin}} $

  1. (15) Find all matrices
    $$
    X=\left(
    \begin{array}{cc}
    x&y\\
    z&w
    \end{array}
    \right)
    $$
    such that $latex XA=BX$, where
    $$
    A=\left(
    \begin{array}{cc}
    1&2\\
    -1&0
    \end{array}
    \right), \mathrm{\ and\ }
    B=\left(
    \begin{array}{cc}
    0&1\\
    3&0
    \end{array}
    \right).
    $$
  2. (15) Same for
    $$
    A=\left(
    \begin{array}{cc}
    2&0\\
    0&0
    \end{array}
    \right), \mathrm{\ and\ }
    B=\left(
    \begin{array}{cc}
    1&3\\
    1&-1
    \end{array}
    \right).
    $$
  3. (20) Find $latex LU$ decomposition for the Vandermonde matrices of sizes 2,3,4:
    $$
    V_2=\left(
    \begin{array}{cc}
    1&1\\
    x_1&x_2
    \end{array}
    \right);
    V_3=\left(
    \begin{array}{ccc}
    1&1&1\\
    x_1&x_2&x_3\\
    x^2_1&x^2_2&x^2_3
    \end{array}
    \right), \mathit{etc.}
    $$
  4. (20) Find matrix $latex A$ (of smallest possible size), such that  $latex A^3\neq 0, A^4=0$.
  5. (30) Find the general formula for the determinant of the tridiagonal $latex n\times n$ matrix
    $$
    \det\left(
    \begin{array}{ccccc}
    3& 2& 0&\ldots\\
    1& 3& 2&\ldots\\
    0& 1& 3&\ldots\\
    \vdots&\vdots&\vdots&\ddots
    \end{array}
    \right)
    $$
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