What is rational canonical form?
Solution
Let \(T:V\to V\) be a linear operator on a finite-dimensional vector space over a field \(F\).
1. Regard \(V\) as an \(F[x]\)-module by letting\[ x\cdot v=T(v). \]
For a polynomial \(f(x)=\sum_i a_ix^i\), define \[ f(x)\cdot v=f(T)v=\sum_i a_iT^i(v). \] This gives an \(F[x]\)-module structure. Since \(V\) is finite-dimensional over \(F\), it is finitely generated as an \(F[x]\)-module. It is torsion because the minimal polynomial \(m_T\) satisfies \(m_T(T)=0\).
2. By the structure theorem for finitely generated torsion modules over the PID \(F[x]\),\[ V\cong \bigoplus_{i=1}^r F[x]/(f_i(x)), \qquad f_1\mid f_2\mid\cdots\mid f_r, \]for uniquely determined monic invariant factors \(f_i\).
The ring \(F[x]\) is a PID. Applying the invariant-factor form of the structure theorem gives the decomposition, with uniqueness of the monic factors subject to the divisibility chain.
3. On a cyclic summand \(F[x]/(f)\), multiplication by \(x\) is represented in the basis\[ 1,x,\ldots,x^{d-1},\qquad d=\deg f, \]by the companion matrix \(C(f)\).
Write \[ f(x)=x^d+a_{d-1}x^{d-1}+\cdots+a_0. \] Multiplication by \(x\) sends \(x^i\) to \(x^{i+1}\) for \(i<d-1\), while in the quotient \[ x^d=-a_0-a_1x-\cdots-a_{d-1}x^{d-1}. \] These are exactly the columns of the companion matrix \(C(f)\).
4. Therefore the matrix of \(T\) is similar to\[ C(f_1)\oplus C(f_2)\oplus\cdots\oplus C(f_r). \]This block diagonal matrix is the rational canonical form of \(T\).
Choose on each cyclic summand the basis from 3 and concatenate these bases. Under the module isomorphism from 2, multiplication by \(x\) corresponds to \(T\), so its matrix is precisely the displayed block diagonal matrix.
5. The rational canonical form is unique up to the order convention on the blocks, and it exists over every field.
The invariant factors \(f_1,\ldots,f_r\) are uniquely determined by the \(F[x]\)-module \(V_T\), so the corresponding companion blocks are uniquely determined. No splitting of the polynomials is required, unlike Jordan form, so rational canonical form exists over an arbitrary field.
6. The largest invariant factor is the minimal polynomial, and the product of the invariant factors is the characteristic polynomial:\[ m_T=f_r, \qquad \chi_T=\prod_{i=1}^r f_i. \]
The annihilator of the direct sum is the least common multiple of the annihilators \((f_i)\), which is \((f_r)\) because of the divisibility chain; hence \(m_T=f_r\). Also the characteristic polynomial of a companion matrix \(C(f_i)\) is \(f_i\), and characteristic polynomials multiply across block diagonal sums.