(Written in my linear algebra book.)
$$A^6 = \left[C\begin{pmatrix} 2&0\\0&3 \end{pmatrix} C^{-1}\right]^6 = C\begin{pmatrix} 2^6 &0\\0& 3^6 \end{pmatrix}C^{-1}$$
But why is it like that? Wouldn't the power affect to the three matrix?
2026-04-03 21:10:30.1775250630
Powers in matrix product
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That's in fact due to matrix decomposition. If a matrix M can be written as $M=CDC^{-1}$ with D being diagonal and C being invertible then for any power $n$ of $M$ we have: $M^{n}=M*M*M*...*M$ (ntimes). Replacing the decomposition we have: $M^{n}=CDC^{-1}CDC^{-1}...CDC^{-1}$. Since $CC^{-1}=I$ by consecutive operation we have $M^{n}=CD^{n}C^{-1}$ in which $D^{n}$ has been defined to be extracted form diagonal $D$ by power affecting on diagonal entries.