There are finitely many \(6 \times 6\) matrices over \(\mathbb{Q}\), in rational canonical form, with minimal polynomial \((x + 2)^2(x - 1)\). Find them.
Solution
Put \(p=x+2\), \(q=x-1\), and \(f=p^2q=x^3+3x^2-4\). For a monic polynomial \(h\), write \(C(h)\) for the matrix of multiplication by \(x\) on \(\mathbf Q[x]/(h)\) in the ordered basis \(1,x,\ldots,x^{\deg h-1}\). The answer consists of the following six block diagonal matrices: \[ \begin{gathered} \operatorname{diag}(C(f),C(f)),\\ \operatorname{diag}(C(p),C(p^2),C(f)),\\ \operatorname{diag}(C(p),C(pq),C(f)),\\ \operatorname{diag}(C(q),C(pq),C(f)),\\ \operatorname{diag}(C(p),C(p),C(p),C(f)),\\ \operatorname{diag}(C(q),C(q),C(q),C(f)). \end{gathered} \] The blocks are explicitly \[ \begin{gathered} C(p)=(-2),\qquad C(q)=(1),\\ C(p^2)=\begin{pmatrix}0&-4\\1&-4\end{pmatrix},\qquad C(pq)=\begin{pmatrix}0&2\\1&-1\end{pmatrix},\qquad C(f)=\begin{pmatrix}0&0&4\\1&0&0\\0&1&-3\end{pmatrix}. \end{gathered} \]
1. The invariant factors form a chain of nonconstant monic polynomials\[ d_1\mid d_2\mid\cdots\mid d_s=f, \qquad \sum_{i=1}^s\deg d_i=6. \]
Apply the rational canonical form theorem to \(\mathbf Q^6\) as a \(\mathbf Q[x]\)-module, with \(x\) acting by the given matrix [@DF04]. Its invariant factors determine the companion blocks, their degrees sum to the dimension, and their largest member is the minimal polynomial. Since \(\deg f=3\), all the earlier invariant factors have total degree \(3\) and divide \(f\).
2. There are exactly six such chains.
The positive degrees before the final factor \(f\) are nondecreasing and sum to \(3\). Their only possibilities are \((3)\), \((1,2)\), and \((1,1,1)\).
For \((3)\), a monic degree-\(3\) divisor of \(f\) is \(f\) itself. This gives \((f,f)\).
For \((1,2)\), the degree-\(2\) factor is either \(p^2\) or \(pq\). If it is \(p^2\), the preceding linear factor must be \(p\). If it is \(pq\), that factor can be either \(p\) or \(q\). Thus the three chains are \((p,p^2,f)\), \((p,pq,f)\), and \((q,pq,f)\).
For \((1,1,1)\), divisibility forces the three monic linear factors to coincide. They are all \(p\) or all \(q\), giving \((p,p,p,f)\) and \((q,q,q,f)\). This exhausts the possible chains.
3. Every displayed matrix has the required minimal polynomial, andno two are similar.
The minimal polynomial of \(C(h)\) is \(h\): a polynomial \(g\) annihilates multiplication by \(x\) on \(\mathbf Q[x]/(h)\) exactly when \(h\mid g\), as can be checked by applying it to \(1\). A polynomial annihilates a block diagonal matrix exactly when it annihilates every block, so its minimal polynomial is the least common multiple of the block minimal polynomials. For each displayed divisibility chain this is \(f\), and the block sizes sum to \(6\). The chains are distinct, so uniqueness of the invariant factors makes the six matrices pairwise nonsimilar [@DF04]. Step 2 proves that no further rational canonical form is possible.