Answer to Question #158368 in Linear Algebra for Sristhi

Question

Find all possible rational canonical form for a 6×6 matrix over R with minimal polynomial m(t)= (t² -2t +3)(t+1)²

Answer 1

Find comparision matrices of invariant factors:

for $t+1$ :

$C_1=[-1]$

for $t^2-2t+3$ :

$C_2=\begin{pmatrix} 0 & -3 \\ 1 & 2 \end{pmatrix}$

for $m(t)= (t^2 -2t +3)(t+1)^2=t^4-t^2+t+3$

$C_4=\begin{pmatrix} 0 & 0&0&-3 \\ 1 & 0&0&-1\\ 0&1&0&1\\ 0&0&1&0 \\ \end{pmatrix}$

for $(t^2 -2t +3)(t+1)=t^3-t^2+t+3$

$C_3=\begin{pmatrix} 0 & 0&-3 \\ 1 & 0&-1\\ 0&1& 1 \\ \end{pmatrix}$

If $m(t)=q^{m_1}_1...q^{m_i}_i$ , then

size of block: $n_j(deg(q_{0j}))$

# blocks: $\frac{dim (Null(q_{0j}(A)))}{deg(q_j)}$

Possible rational canonical form for a 6×6 matrix: