Homework 18: Eigenvectors, Eigenvalues, and Eigenspaces

Instructions. Problem 4 is optional.

1. Section 5.1, #5(a)(b)(d), 7, 21, 25(a)(c).
   For Problem 21, there is no need to refer to any tables, as the instructions suggest, and calculations are not required.
2. The vector \(\bf{x}=\left[ \begin{matrix} \pe 1\\ -2 \\ \pe 1 \end{matrix} \right] \) is an eigenvector for \(A= \left[ \begin{matrix} 3&6&7\\ 3&3&7\\ 5&6&5 \end{matrix} \right]\).
   1. Find the eigenvalue \(\lambda\) which corresponds to \(\bf{x}\). Show your work.
   2. Let \(\lambda\) be the eigenvalue you found in (a). Is it possible to find another eigenvector \(\bf{y}\) corresponding to eigenvalue \(\lambda\) such that \(\{\bf{x},\bf{y}\}\) is linearly independent? Justify.
      Hint: Find the dimension of the \(\lambda\)-eigenspace of \(A\).
3. Let \(A= \left[ \begin{matrix} 1&1\\ 4&1 \end{matrix} \right]\), and let \(T:\bR^2\to \bR^2\) be given by \(T(\bx)=A\bx\).
   1. Find all eigenvalues of \(A\).
   2. Find a basis \(B\) such that \([T]_B\) is diagonal.
   3. Find \([T]_B\).
   4. Express \(A\) in the form \(A=PDP^{-1}\) for some diagonal matrix \(D\) and invertible matrix \(P\).
4. (Optional) Section 5.2, #15, 27.