Linear Algebra with Applications, 10th Edition

Section 8.2 Exercises

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

2. Let A be a $5\times 5$ matrix. If $A^2\ne 0$ and $A^3=0$, what are the possible Jordan canonical forms for A?

3. Find the Jordan canonical form J for each of the following matrices and determine a matrix X such that $X^{-1}\mathit{\text{AX}}=J$:

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

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

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

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

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

4. Let L be a linear operator on a finite dimensional vector space V.

   1. Show that $R\left(L^i\right)\subset R\left(L^j\right)$ whenever $i>j$.

   2. If for some $k_0,R\left(L^{k0}\right)=R\left(L^{k_0+1}\right)$, then $R\left(L^{k0}\right)=R\left(L^{k_0+k}\right)$ for all $k\ge 1$.

5. Let L be as in Exercise 4.

   1. Show that there is a smallest positive integer $k_0$ such that $R\left(L^{k_0}\right)=R\left(L^{k_0+1}\right)$.

   2. Let $k_1$ be the smallest ...