Show $ \begin{bmatrix} a & -b \\ b & a \end{bmatrix}^k = \begin{bmatrix} Re((a+ib)^k) & -Im((a+ib)^k) \\ Im((a+ib)^k) & Re((a+ib)^k) \end{bmatrix}$

Question

Let

$A = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$

Claim:

$A^k = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}^k = \begin{bmatrix} Re((a+ib)^k) & -Im((a+ib)^k) \\ Im((a+ib)^k) & Re((a+ib)^k) \end{bmatrix}$

I can see that for $k = 1$ this statement is trivially true

For $k = 2$:

$A^2 = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}^2 = \begin{bmatrix} a^2 - b^2 & -2ab \\ 2ab & a^2-b^2 \end{bmatrix} = \begin{bmatrix} Re((a+ib)^2) & -Im((a+ib)^2) \\ Im((a+ib)^2) & Re((a+ib)^2 \end{bmatrix}$

But this quickly get out of hand and becomes difficult to check for large $k$

What is the trick that carries the LHS expression to the RHS?

Answer 1

You have proved it for $k=2$. Let's suppose that it holds for $k$, we'll show it holds for $k+1$. Let $q=(a+bi)^k$. Note first that $$(a+ib)^{k+1}=q^k(a+bi)=aRe(q)-bIm(q)+(aIm(q)+bRe(q))i $$ Now: $A^{k+1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}^{k+1}=\begin{bmatrix} a & -b \\ b & a \end{bmatrix}^{k}\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$.

By induction hypothesis: $$\begin{bmatrix} a & -b \\ b & a \end{bmatrix}^{k}=\begin{bmatrix} Re(q) & -Im(q) \\ Im(q) & Re(q) \end{bmatrix}$$ So: $$A^{k+1}=\begin{bmatrix} Re(q) & -Im(q) \\ Im(q) & Re(q) \end{bmatrix}\begin{bmatrix} a & -b \\ b & a \end{bmatrix}=\begin{bmatrix} aRe(q)-bIm(q) & -bRe(q)-aIm(q) \\ aIm(q)+bRe(q) & aRe(q)-bIm(q) \end{bmatrix}$$

Which is what we wanted to prove (Why?).

Answer 2

There is a field isomorphism $\eta :\Bbb C\to F$ where $F$ is the set of matrices $M(a,b)= \begin{pmatrix} a&b\\-b&a\end{pmatrix},a,b\in\Bbb R$ under usual matrix sum and multiplication that sends $a+bi$ to $M(a,b)$ and your claim follows from this.

Answer 3

Hint:

The easier way is to prove that $$J=\begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}$$ has the property $J^2=-I$ and you are trying to prove that $A=aI+bJ$ is "like" $a+bi$. Then the induction step is much cleaner.