diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -1816,14 +1816,16 @@ the primary tool we'll use to generalize the results of the previous section.
What is simultaneous diagonalization all about then?
\begin{definition}\label{def:sim-diag}
- A set of square matrices \(\{X_i\}_i\) is called \emph{simultaneously
- diagonalizable} if there some invertible matrix \(P\) such that \(P X_i
- P^{-1}\) is diagonal for every \(i\).
+ Given a \(K\)-vector space \(V\), a set of operators \(\{T_j : V \to V\}_j\)
+ is called \emph{simultaneously diagonalizable} if there is a basis \(\{v_1,
+ \ldots, v_n\}\) for \(V\) such that \(T_j v_i\) is a scalar multiple of
+ \(v_i\), for all \(i, j\).
\end{definition}
\begin{proposition}
- A set of diagonalible matrices is simultaneously diagonalizable if, and only
- if all of its elements commute with one another.
+ Given a \emph{finite-dimensional} vector space \(V\), A set of diagonalizable
+ operators \(V \to V\) is simultaneously diagonalizable if, and only if all of
+ its elements commute with one another.
\end{proposition}
\begin{corollary}
@@ -1843,19 +1845,7 @@ What is simultaneous diagonalization all about then?
% TODOO: Prove that the operators are diagonalizable
\begin{proof}
It suffices to show that \(H : V \to V\) is a diagonalizable operator for
- each \(H \in \mathfrak{h}\). Indeed, if this is the case it follows from the
- fact that \(\mathfrak{h}\) is Abelian that the set
- \(\{[H\!\restriction_V]_{\mathcal{B}} : H \in \mathfrak{h}\}\) is
- simultaneously diagonalible for any basis \(\mathcal{B}\) for \(V\).
-
- If \(P\) is as in definition~\ref{def:sim-diag} and \(T\) is the operator \(V
- \to V\) such that \([T]_{\mathcal{B}}\), then \(\mathcal{C} =
- T(\mathcal{B})\) is a basis for \(V\) such that
- \([H\!\restriction_V]_{\mathcal{C}} = P [H\!\restriction_V]_{\mathcal{B}}
- P^{-1}\) is diagonal for each \(H \in \mathfrak{h}\). If we then take
- \(\{v_1, \ldots, v_n\} = \mathcal{C}\) and denote by \(\lambda_i(H)\) the
- coefficient of \(E_{i i}\) in \([H\!\restriction_V]_{\mathcal{C}}\), we find
- \(H v_i = \lambda_i(H) \cdot v_i\) as intended.
+ each \(H \in \mathfrak{h}\).
\end{proof}
As promised, this implies\dots