For an operator \(T \in \mathcal{L}(V)\), \(T^{n}\) would make sense. Instead of writing \(TTT\dots\), then, we just write \(T^{n}\).
constituents
- operator \(T \in \mathcal{L}(V)\)
requirements
- \(T^{m} = T \dots T\)
additional information
\(T^{0}\)
\begin{equation} T^{0} := I \in \mathcal{L}(V) \end{equation}
\(T^{-1}\)
\begin{equation} T^{-m} = (T^{-1})^{m} \end{equation}
if \(T\) is invertable
usual rules of squaring
\begin{equation} \begin{cases} T^{m}T^{n} = T^{m+n} \\ (T^{m})^{n} = T^{mn} \end{cases} \end{equation}
This can be shown by counting the number of times \(T\) is repeated by writing each \(T^{m}\) out.