Linear Algebra with Applications, 10th Edition

Section 8.1 Exercises

1. Let L be a linear operator on a vector space V of dimension 5 and let A be any matrix representing L. If L is nilpotent of index 3, then what are the possible Jordan canonical forms of A?

2. Let A be a $4\times 4$ matrix whose only eigenvalue is $\lambda =2$. What are the possible Jordan canonical forms of A?

3. Let L be a linear operator on a vector space V of dimension 6 and let A be a matrix representing L. If L has only one distinct eigenvalue $\lambda$ and the eigenspace $S_{\lambda }$ has dimension 3, then what are the possible Jordan canonical forms of A?

4. For each of the following, find a matrix S such that $S^{-1}\mathit{\text{AS}}$ is a simple Jordan matrix:

   1. $A=\left[\begin{array}{ccc}1& 0& 1\\ 1& 0& 2\\ 1& -1& 2\end{array}\right]$

   2. $A=\left[\begin{array}{cccc}1& 2& 0& 0\\ 0& 1& 2& 0\\ 0& 0& 1& 2\\ 0& 0& 0& 1\end{array}\right]$

5. For each of the following, find a matrix S such that $S^{-1}\mathit{\text{AS}}$ is the Jordan canonical ...