Give the line of reflection or angle of rotation of an orthogonal 2x2 matrix

Question

Determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of reflection.

\begin{bmatrix}-\frac{3}{5}&-\frac{4}{5}\\-\frac{4}{5}&\frac{3}{5}\end{bmatrix}

Since it is of the form (det(A) = -1):

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

I figured it must be a reflection. However I have no clue how to find the reflection line. This question Finding reflection line or surface from reflection matrix gives solutions to a similar problem however it uses eigenvalues. We haven't touched the subject of eigenvalues (yet) so I'm wondering if I can solve this problem without eigenvalues.

Answer 1

If it is a reflection, there must be a vector which is not changed by this transformation. That is the vector you are looking for.

Or, in short, solve $$\left[\begin{matrix}-3/5 & -4/5 \\ -4/5 & 3/5\end{matrix}\right]\vec{v}=\vec{v}$$

Answer 2

If $A$ reflects through the line in the direction of $x$, then $Ax=x$. Solve this equation for $x$.

Answer 3

Take any point, $p$, and compute $q= Ap$. The mirror axis is the perpendicular bisector of $pq$. To make the arithmetic easy, we may as well take $p=(1,0)^T$. Then $q= (-3/5, -4/5)$, so $p-q = (8/5, 4/5)$, which is in the direction $(2,1)$. The mirror axis is at right angles to this, so it's in the direction $(-1,2)$.

Though you may not realize it, the other two answers are using eigenvalues (but without mentioning the word).