Section 7.4 Triangulable linear maps and matrices
The matrix form of the above definition is given below.
Definition 7.4.2. (Triangulable matrix).
A matrix \(A\in M_n(F)\) is said to be triangulable if \(A\) is similar to an upper triangular matrix in \(M_n(F)\text{.}\)We now state the useful observation.
Proposition 7.4.3.
Let \(V\) be a finite-dimensional vector space over a field \(F\) and let \(T\colon V\to V\) is an \(F\)-linear map. Suppose that \(W\) is a \(T\)-invariant subspace of \(T\text{.}\) Let \(\overline{T}\colon V/W\to V/W\) be an \(F\)-linear map given by \(v+W\mapsto T(v)+W\text{.}\) Then
\begin{equation*}
\chi_T=\chi_{T|_W}\cdot\chi_{\overline{T}}\quad\text{and}\quad tr(T)=tr(T|_W)+tr(\overline{T}).
\end{equation*}
Proof.
We now give a criterion for a linear to be triangulable in terms of its characteristic polynomial. For this first recall the definition of a split polynomial Definition A.1.7.
Theorem 7.4.4.
Let \(V\) be a finite-dimensional vector space of dimension \(n\) over a field \(F\) and \(T\colon V\to V\) be an \(F\)-linear map. The map \(T\) is triangulable if and only if the characteristic polynomial of \(T\text{,}\) \(\chi_T\) is split over \(F\text{.}\)Proof.
Suppose that \(T\) is triangulable, i.e., there is a basis \(\mathfrak{B}\) of \(V\) in which \([T]_{\mathfrak{B}}=(a_{ij})\) is an upper triangular matrix, i.e., \(a_{ij}=0\) for \(i>j\text{.}\) Thus \(\chi_T=\det(tI_n-[T]_{\mathfrak{B}})=(t-a_{11})\cdots(t-a_{nn})\) by a property of the determinant, see Section A.2. Hence \(\chi_T\) is split over \(F\text{.}\)
Now assume that \(\chi_T\) is split over \(F\text{.}\) We show that \(T\) is triangulable. We proceed by induction on the dimension of \(V\text{,}\) \(\dim_FV=n\text{.}\) If \(n=1\) then the result is clear. Assume that \(n>1\text{.}\) Since \(\chi_T\) is split, it has a root \(\lambda\in F\text{,}\) i.e., \(\lambda\in F\) is an eigenvalue of \(T\text{.}\) Let \(v_1\) be an eigenvector corresponding to \(\lambda\in F\text{.}\) Thus \(W=\langle v_1\rangle\) is an invariant subspace under \(T\text{.}\) We define a linear map \(\overline{T}\colon V/W\to V/W\) by
\begin{equation*}
v+W\mapsto T(v)+W.
\end{equation*}
By Proposition 7.4.3 we have \(\chi_T=\chi_{T|_W}\cdot\chi_{\overline{T}}\text{.}\) Therefore, \(\chi_{\overline{T}}\) is also split. As \(\dim_FV/W < n\text{,}\) by induction, there is a basis \(\overline{\mathfrak{B}}=\{v_2+W,\ldots,v_n+W\}\) of \(V/W\) such that \([\overline{T}]_{\overline{\mathfrak{B}}}\) is an upper triangular matrix. Then for the basis \(\{v_1,v_2,\ldots,v_n\}\) of \(V\text{,}\) the matrix of \(T\) is upper triangular.
