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@jemoka / Jemoka Knowledge Base / wiki/concepts/raising_operators_to_powers.md
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--- title: "T^m" type: concept related: [Invertability, Operator] source: https://www.jemoka.com/posts/kbhraising_operators_to_powers/ confidence: high status: active --- 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.