Section2.21PID-module structure theorems: Jordan canonical forms
Amazingly, by treating the case \(R=F[x]\text{,}\) where \(F\) is a field, the PID-module structure theorems gives us some advanced theory about linear transformations for finite-dimensional \(F\)-vector spaces essentially for free! Before getting into details, we first make an observation about modules over a general polynomial ring.
If \(M\) is an \(R[x]\)-module, then in particular \(M\) is an \(R\)-module, and the map \(\phi_x\colon M\rightarrow M\) defined as \(\phi_x(m)=x\cdot m\) is an \(R\)-module endomorphism of \(M\text{.}\)
Conversely, if \(M\) is an \(R\)-module and \(\phi\colon M\rightarrow M\) is an \(R\)-module endomorphism of \(M\text{,}\) then \(M\) can be a given an \(R[x]\)-module structure by defining
\begin{align*}
(\sum_{k=0}^na_kx^k) \cdot m \amp = \sum_{k=0}^na_k\phi^k(m) \text{.}
\end{align*}
In other words, an \(R[x]\)-module consists of an \(R\)-module \(M\text{,}\) together with a choice of \(\phi\in \End_R(M)\) that determines how elements of \(M\) are multiplied by \(x\text{.}\)
The proposition is the key that will allow us to connect linear transformations of vector spaces to the PID-module structure theory. Firstly, if \(F\) is a field, then we know that \(F[x]\) is a PID. Now, let \(V\) be a finite-dimensional \(F\)-vector space, and let \(T\colon V\rightarrow V\) be a linear transformation of \(V\text{.}\) Using the language of module theorem, \(V\) is a finite free \(F\)-module, and \(T\in \End_F(V)\) is an \(F\)-endomorphism of \(V\text{.}\)
According to our proposition, \(V\) inherits an \(F[x]\)-module structure from \(T\) by defining \(x^n\cdot v=T^n(v)\text{,}\) and more generally, \(g(x)\cdot v=g(T)(v)\text{.}\) Since \(V\) is finitely generated over \(F\text{,}\) it is certainly finitely generated over \(F[x]\text{.}\) We conclude that as an \(F[x]\)-module, we have
\begin{align*}
V \amp \cong (F[x])^n\oplus\bigoplus_{i=1}^m F[x]/(g_i(x))
\end{align*}
for some polynomials \(g_i(x)\in F[x]\text{.}\) Next, since \(V\) is a finite-dimensional \(F\)-vector space and \(F[x]\) is an infinite-dimensional \(F\)-vector space, we must have \(n=0\) and
\begin{align*}
V \amp \cong \bigoplus_{i=1}^m F[x]/(g_i(x)) \text{.}
\end{align*}
Great, but what does this tell us about the linear transformation \(T\text{?}\)
Remark2.21.3.Minimal and characteristic polynomial.
Assume \(V\) is a \(n\)-dimensional \(F\)-vector space, and let \(T\in \End_F(V)\) be a linear transformation. The minimal polynomial is the unique monic generator of
\begin{align*}
\Ann V \amp =\{g\in F[x]\mid g(T)=0_V\}\text{.}
\end{align*}
By the Cayley-Hamilton theorem (proved in great generality in your homework), we have \(\ch_T(T)=0_V\text{,}\) and thus \(m_T(x)\mid \ch_T(x)\text{.}\)
There exists a unique set of distinct monic irreducible polynomials \(p_1(x),p_2(x),\dots, p_r(x)\text{,}\) and for each \(1\leq i\leq r\) a unique sequence of positive integers
Let \(V\) be an \(n\)-dimensional \(F\)-vector space, and let \(T\in \End_F(V)\) be a nonzero linear transformation. We treat \(V\) as an \(F[x]\)-module via the action of \(T\) on \(V\text{.}\)
Let \((a_i(x))_{i=1}^m\) be the monic invariant factors of \(V\text{.}\) We have
Let \(p_1(x),p_2(x),\dots, p_r(x)\) be the distinct monic irreducible polynomials appearing in the elementary factors decomposition of \(V\text{,}\) and for each \(1\leq i\leq r\text{,}\) let \((n_{ij})_j\) be the accompanying unique sequence of integers. We have
