Let \(A\) be an \(n\times n\) matrix over a field \(k\). The rational canonical form of \(A\) is the block diagonal matrix of the companion matrices of the invariant factors of the \(k[x]\)-module \(k^n\), with \(x\) acting by \(A\). It exists over every field; the Jordan form exists if and only if \(\chi_A\) splits over \(k\).
Companion matrix (Tag D-HJR7M)
Let \(k\) be a field and let \(p(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1} + x^n\in k[x]\) be a monic polynomial of degree \(n\geq 1\). The companion matrix of \(p\) is \[ C_p \coloneqq \begin{bmatrix} 0 & 0 & \cdots & 0 &-a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1} \end{bmatrix} \in\Mat_{n\times n}(k). \]
Rational canonical form (Tag FD-XT6HD)
Let \(k\) be a field, let \(V\) be a finite-dimensional \(k\)-vector space, and let \(\phi\colon V\to V\) be a \(k\)-linear map. Make \(V\) a \(k[x]\)-module by letting \(x\) act as \(\phi\). By the structure theorem for finitely generated modules over the principal ideal domain \(k[x]\), there are unique monic nonconstant polynomials \(r_1\divides r_2\divides\cdots\divides r_t\) in \(k[x]\) and a decomposition \(V=V_1\oplus\cdots\oplus V_t\) into \(\phi\)-invariant subspaces with \[ V_i\cong k[x]/(r_i)\quad\text{as } k[x]\text{-modules},\qquad 1\le i\le t. \] The rational canonical form of \(\phi\) is the block diagonal matrix \[ \diag\qty{C_{r_1},C_{r_2},\ldots,C_{r_t}}, \] where \(C_{r_i}\) is the companion matrix of \(r_i\).
The polynomial \(r_i\) is the minimal polynomial of the restriction \(\phi|_{V_i}\). If \(v_i\in V_i\) corresponds to the class of \(1\) in \(k[x]/(r_i)\) and \(d_i=\deg r_i\), then \(v_i,\phi(v_i),\ldots,\phi^{d_i-1}(v_i)\) is a basis of \(V_i\), and the matrix of \(\phi|_{V_i}\) in this basis is \(C_{r_i}\). Concatenating these bases gives a basis of \(V\) in which \(\phi\) has its rational canonical form.
Equivalent conditions for an operator to have a cyclic vector (Tag PR-4GQIZ)
Let \(k\) be a field, let \(V\) be a nonzero finite-dimensional \(k\)-vector space, and let \(T\colon V \to V\) be a \(k\)-linear map. Make \(V\) a \(k[x]\)-module by \(f(x) \cdot v \coloneqq f(T)v\), and let \(\min_T(x)\) be the minimal polynomial of \(T\). The following are equivalent:
\(V \cong k[x]/\gens{\min_T}\) as \(k[x]\)-modules.
\(V\) admits a cyclic vector: there is \(v \in V\) such that \(v, Tv, T^2v, \ldots\) span \(V\). For such \(v\), \(\min_T\) is the monic polynomial \(p\) of least degree with \(p(T)v = 0\).
\(V\) is a cyclic \(k[x]\)-module. Its annihilator is then the ideal \(\gens{\min_T}\).
The matrix of \(T\) in some basis of \(V\) is the companion matrix \(C_{\min_T}\).
\(\min_T(x) = \det(xI - T)\), the monic characteristic polynomial of \(T\).
\(V\) has exactly one invariant factor as a \(k[x]\)-module.
The rational canonical form of \(T\) has a single block.
The minimal and characteristic polynomials agree iff there is a cyclic vector (Tag PR-GBL6P)
Let \(k\) be a field and \(A \in \Mat_{n\times n}(k)\) with \(n \geq 1\), and let \(\min_A(x)\) be the minimal polynomial of \(A\). Then \(\min_A(x) = \det(xI - A)\) if and only if \(A\) admits a cyclic vector: some \(v \in k^n\) such that \(v, Av, A^2v, \ldots\) span \(k^n\).
Let \(V = k^n\) with \(x\) acting by \(A\).
\(\impliedby\): Suppose \(v\) is a cyclic vector, so that \(v, Av, \ldots, A^{n-1}v\) is a basis of \(V\). If \(n' \da \deg\min_A < n\), then \(\min_A(A)v = 0\) is a linear dependence among \(v, Av, \ldots, A^{n'}v\), a contradiction. Hence \(\deg\min_A = n = \deg\chi_A\), and since \(\min_A\) divides \(\chi_A\) and both are monic, \(\min_A = \chi_A\).
\(\implies\): By the structure theorem \(V\cong \bigoplus_{i=1}^m k[x]/\gens{p_i}\) with nonconstant invariant factors \(p_1\divides\cdots\divides p_m\), \(\chi_A = \prod_i p_i\), and \(\min_A = p_m\). If \(\min_A = \chi_A\), comparing degrees gives \(m = 1\), so \(V\cong k[x]/\gens{p_m}\) is a cyclic \(k[x]\)-module. If \(v\) generates \(V\), then \(V = \ts{f(A)v \st f\in k[x]}\) is spanned by \(v, Av, A^2v, \ldots\), and by Cayley–Hamilton by \(v, Av, \ldots, A^{n-1}v\).
Rational canonical form (Tag PR-TI6YA)
Let \(k\) be a field and \(A\in M_n(k)\), and regard \(k^n\) as a \(k[x]\)-module with \(x\) acting by \(A\). Let \(r_1\divides r_2\divides\cdots\divides r_m\) be the monic nonconstant invariant factors of this module. Then \(A\) is similar to the block diagonal matrix \[ \RCF(A)=\diag(C_{r_1},\ldots,C_{r_m}), \] where \(C_{r_i}\) is the companion matrix of \(r_i\), and \(\RCF(A)\) is the unique matrix of this form similar to \(A\).
By the structure theorem for finitely generated modules over the PID \(k[x]\), \(k^n\cong\bigoplus_{i=1}^m k[x]/(r_i)\) as \(k[x]\)-modules, with the monic \(r_i\) uniquely determined. On \(k[x]/(r_i)\), multiplication by \(x\) in the basis \(1,x,\ldots,x^{\deg r_i-1}\) has matrix \(C_{r_i}\). Two matrices are similar if and only if the corresponding \(k[x]\)-modules are isomorphic, so uniqueness of the invariant factors gives uniqueness of \(\RCF(A)\).
The blocks of the rational canonical form of \(A\) are in bijection with the nonconstant invariant factors of \(xI-A\), and a companion matrix is its own rational canonical form.
Derivation
(Derivation of the rational canonical form)
Let \(T\colon V\to V\) be the linear map given by \(A\), and let \(k[x]\) act on \(V\) by \(p(x)\actson \vector v \da p(T)\vector v\). This makes \(V\) a finitely generated torsion \(k[x]\)-module whose submodules are the \(T\)-invariant subspaces.
Since \(k[x]\) is a PID, the structure theorem gives the invariant factor decomposition \[ V \cong \bigoplus_{i=1}^m k[x] / \gens{ p_i(x) }, \qquad p_1 \divides p_2 \divides \cdots \divides p_m. \]
Each summand is a \(T\)-invariant subspace \(V_i\) on which the minimal polynomial of \(T\) is \(p_i\). Since \(p_i \divides p_m\) for every \(i\), \(p_m(T) = 0\) on \(V\), so \(\min_T \divides p_m\); since \(\min_T(T)\) vanishes on \(V_m\cong k[x]/\gens{p_m}\), \(p_m\divides\min_T\). Hence \(\min_T = p_m\).
\(T\) is block diagonal with one block for each \(V_i\), so it suffices to treat a cyclic module \(V = k[x]/\gens{p(x)}\) with \(p(x) = x^n + a_{n-1}x^{n-1}+\cdots+a_0\) and generator \(\vector v\).
Then \(\vector v, T\vector v, \ldots, T^{n-1}\vector v\) is a basis: a linear dependence among them would give a nonzero polynomial of degree less than \(n\) annihilating the generator \(\vector v\), hence annihilating \(V\), contradicting that \(p\) is the minimal polynomial of \(T\).
In this basis \(T\) sends each basis vector to the next, and \(T^n\vector v = -\sum_{i<n}a_iT^i\vector v\) because \(p(T)=0\), so the matrix of \(T\) is the companion matrix \[ C(p) = \begin{bmatrix} 0 & & & & -a_0 \\ 1 & 0 & & & -a_1 \\ & 1 & 0& & -a_2 \\ & & \ddots & \ddots & \vdots \\ & & & 1 & -a_{n-1} \end{bmatrix}, \] whose characteristic polynomial is \(p\). Hence \(\chi_T = \prod_i p_i\).
Cyclic vectors
Equivalent characterizations of a single nilpotent Jordan block of size $n$ (Tag PR-K6MMW)
Let \(k\) be a field, \(V\) a \(k\)-vector space of dimension \(n \coloneqq \dim_k V \geq 1\), and \(T\colon V\to V\) a nilpotent linear map. The following are equivalent:
There exists a basis \(e_1, \ldots, e_n\) of \(V\) such that \[ T(e_i) = \begin{cases} e_{i-1} & \text{if } i \geq 2, \\ 0 & \text{if } i=1. \end{cases} \]
There exists a cyclic vector \(v \in V\): the vectors \(T^k v\) for \(k = 0, 1, \ldots, n-1\) form a basis of \(V\).
\(T^{n-1} \neq 0\).
\(\dim_k \ker T^\ell = \ell\) for each \(1\leq \ell \leq n\).
\(\dim_k \ker T = 1\).