I find something confusing. Consider the matrix $\begin{bmatrix}a & -b \\ b & a\end{bmatrix}$, we can find $\begin{bmatrix}a & -b \\ b & a\end{bmatrix}^{k} = \begin{bmatrix}Re(\lambda^k) & -Im(\lambda^k) \\ Im(\lambda^k) & Re(\lambda^k)\end{bmatrix}$ by induction, where $\lambda = a + ib$.
I am a little confused about this interesting result. Is there any theorem behind such phenomenon?
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If you write $a+bi$ as $\rho\bigl(\cos(\theta)+\sin(\theta)i\bigr)$ (with $\rho\geqslant0$), then$$\begin{bmatrix}a&-b\\b&a\end{bmatrix}=\rho\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$and therefore$$\begin{bmatrix}a&-b\\b&a\end{bmatrix}^k=\rho^k\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}^k=\rho^k\begin{bmatrix}\cos(k\theta)&-\sin(k\theta)\\\sin(k\theta)&\cos(k\theta)\end{bmatrix}.$$Now, use the fact that $(a+bi)^k=\rho^k\bigl(\cos(k\theta)+\sin(k\theta)i\bigr)$.
Yes. We have an injective $\mathbb{R}$-algebra morphism $$f:\lambda =a+ib\in\mathbb{C}\mapsto \pmatrix{a & -b\cr b & \phantom{-}a}\in M_2(\mathbb{R}).$$
Injectivity is clear, and the fact that $f$ respects addition and multiplication, as well as $\mathbb{R}$-linearity, are just simple computations.
In particular, we have $f(\lambda^k)=f(\lambda)^k$ for all $\lambda\in\mathbb{C}$ and all $k\geq 0$. Using the definition of $f$, this is exactly the equality you seek.