Question 7.5.7: Let A be a real 2 × 2 matrix with eigenvalues a ± ib (where ......
Let A be a real 2 × 2 matrix with eigenvalues a ± ib (where b ≠ 0). Show that A is similar (over ℝ) to the matrix \begin{bmatrix}a&−b\\b&a\end{bmatrix}, representing a rotation combined with a scaling.
Let \vec{v} ± i\vec{w} be eigenvectors of A with eigenvalues a ± ib. See Exercise 42. Matrix A is similar to \begin{bmatrix}a + ib&0\\0&a − ib\end{bmatrix}; more precisely,
\begin{bmatrix}a + ib&0\\0&a − ib\end{bmatrix}=P^{-1}AP,
where P=\begin{bmatrix}\vec{v}+i\vec{w}&\vec{v}-i\vec{w}\end{bmatrix}. By Example 6, matrix
\begin{bmatrix}a&-b\\b&a\end{bmatrix}
is similar to
\begin{bmatrix}a + ib&0\\0&a − ib\end{bmatrix}
as well, with
\begin{bmatrix}a + ib&0\\0&a − ib\end{bmatrix}= R^{-1}\begin{bmatrix}a&-b\\b&a\end{bmatrix}R, where R=\begin{bmatrix}i&-i\\1&1\end{bmatrix}.
Thus,
P^{-1}AP=R^{-1}\begin{bmatrix}a&-b\\b&a\end{bmatrix}R,
and
\begin{bmatrix}a&-b\\b&a\end{bmatrix}=RP^{-1}APR^{-1}=S^{-1}AS,
where S=PR^{-1} and S^{-1}=(PR^{-1})^{-1}=RP^{-1}.
A straightforward computation shows that
S = P R^{-1}=\frac{1}{2i}\begin{bmatrix}\vec{v}+i\vec{w}&\vec{v}-i\vec{w}\end{bmatrix}\begin{bmatrix}1&i\\-1&i\end{bmatrix}=\begin{bmatrix}\vec{w}&\vec{v}\end{bmatrix};
note that S has real entries, as claimed.