Math 6610 • Analysis of Numerical Methods I

{The Companion Matrix} Show that
\[
  \begin{aligned}
  &\det\left(
    \begin{bmatrix}
      \alpha_{n-1} & \alpha_{n-2} & \cdots & \alpha_1 & \alpha_0\\
      1 & 0 & \cdots & 0 & 0\\
      0 & 1 & \cdots & 0 & 0\\
      \vdots & \vdots & \ddots & \vdots & \vdots\\
      0 & 0 & \cdots & 1 & 0
    \end{bmatrix}
    - \lambda I
  \right) \\
  &\qquad =
  (-1)^n\left[\lambda^n-\sum_{j=0}^{n-1}\alpha_j\lambda^j\right].
  \end{aligned}
  \tag{1}
\]
Describe how this fact can be used to find the roots of a polynomial. Also
briefly describe why the reverse process, finding the eigenvalues of a matrix by
computing the characteristic polynomial, and finding its roots, is not generally
viable.