Homework 8

Consider the two matrices
$A=\left[\begin{array}{ccc}0& -4& 85\\ 1& 4& -30\\ 0& 0& 3\end{array}\right],B=\left[\begin{array}{ccc}2& 2& 1\\ 0& 2& -1\\ 0& 0& 3\end{array}\right].$

1. Show that $A$ and $B$ have the same characteristic polynomial, namely $c_A(x)=c_B(x)=(x-2{)}^2(x-3)$.
2. Show that $(A-2I)(A-3I)\ne 0$ and $(B-2I)(B-3I)\ne 0$, and conclude that $A$ and $B$ have the same minimal polynomial, namely $m_A(x)=m_B(x)=(x-2{)}^2(x-3)$.
3. Show that $A$ and $B$ have the same invariant factor(s) and hence same rational canonical form. Write down their shared rational canonical form matrix.

1. Suppose $A$ and $B$ are non-scalar $2\times 2$ matrices over a field $F$; i.e., neither $A$ nor $B$ is a scalar multiple of the identity matrix. Prove that $A$ and $B$ are similar if and only if they have the same characteristic polynomial.
2. Suppose $A$ and $B$ are $3\times 3$ matrices over a field $F$. Prove that $A$ and $B$ are similar if and only if they have the same characteristic and same minimal polynomials.
3. Give an example of a pair of $4\times 4$ matrices $A$ and $B$ that have the same characteristic and minimal polynomials but are not similar.

Let $A$ be a $3\times 3$ matrix over the field $\mathbf{Q}$.

1. Show that $A^6=I_3$ if and only if the minimal polynomial of $A$ divides $x^6-1$ in $\mathbf{Q}[x]$.
2. The irreducible factorization of $x^6-1$ in $\mathbf{Q}[x]$ is
   $x^6-1=(x-1)(x+1)(x^2-x+1)(x^2+x+1).$
   Show that if $A^6=I_3$ then there are at most nine possibilities for the minimal polynomial of $A$.
3. Continuing part (2), show that there are exactly eight possible lists of invariant factors for such a matrix $A$.
4. Use part (3) to write down all elements of order 1, 2, 3, and 6 in the group ${\mathrm{GL}}_3(\mathbf{Q})$, up to similarity.

Let $A$ be the $4\times 4$ matrix
$A=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ -2& -2& 0& 1\\ -2& 0& -1& -2\end{array}\right].$

1. Show that the invariant factors of $A$ are $a_1(x)=x-1$, $a_2(x)=(x-1)(x+1{)}^2$.
2. Determine the rational canonical form $R$ of $A$ and find a change-of-basis matrix $P$ such that $P^{-1}AP=R$.
3. Determine the Jordan canonical form $J$ of $A$ and find a change-of-basis matrix $Q$ such that $Q^{-1}AQ=J$.

Find all possible Jordan canonical forms of $5\times 5$ matrices over $\mathbf{Q}$ with characteristic polynomial $c(x)=(x-3{)}^2(x-2{)}^3$.

Find all similarity classes of $6\times 6$ matrices over $\mathbf{C}$ with characteristic polynomial $c(x)=(x^2-1)(x^4-1)$.

Find all similarity classes of $6\times 6$ matrices over $\mathbf{Q}$ with minimal polynomial $m(x)=(x-1)(x+2{)}^2$.