Direction of a shear confusion I have this matrix which has a line of invariant points $y=0.5x$ and a line of invariance $y=0.25x$. This matrix is:
\begin{bmatrix}
-3&8\\
-1&3\\
\end{bmatrix}
I am further told that it is a combination of a Shear and a reflection in the line $y=0.5x$. My trouble arises when I am asked to give a geometrical description of the sheer which I calculated to be:
$0.5\begin{bmatrix}
-13&36\\
-9&23
\end{bmatrix}$
The answer states that this is a sheer that is in the direction parallel to the line $y=0.5x$. I am confused as to how this sheer direction is determined? How does the line of invariant points relate to the direction of the shear in a matrix, can anyone convince me?
EDIT: I have made a realization with the aid of a drawing, I realized that given $y=0.5x$ is a line of invariant points, the position of the points after the matrix will lie along that line hence moving parallel to the line $y=0.5x$.
 A: A shear in two dimensions leaves points along some line fixed and shifts other points in a direction parallel to this line proportionally to their distance from the fixed line. In this case, the fixed line is the “line of invariant points” $2x=y$, so the shear is indeed parallel to this line.  
A pure shear matrix only has the one fixed line, but this transformation also maps the line $4x=y$ to itself, so there’s a bit more going on. This line can be represented parametrically as $(4t,t)$ and from $$\begin{bmatrix}-3&8\\-1&3\end{bmatrix}\begin{bmatrix}4t\\t\end{bmatrix}=\begin{bmatrix}-4t\\-t\end{bmatrix}$$ we see that the transformation maps points along this line to their reflections in the origin. I’d hesitate to describe this as a reflection relative to the line $2x=y$, though, because that description usually applies to a transformation that reverses the line perpendicular to $2x=y$. I would instead describe this transformation as decomposing a vector into components in the directions of $(2,1)$ and $(4,1)$ and then reversing the latter component.  
Incidentally, the question of which lines are mapped to themselves by a linear transformation is one of the motivations for the idea of eigenvectors and eigenvalues.
A: I have made a realization with the aid of a drawing, I realized that given $y=0.5x$ is a line of invariant points, the position of the points after the matrix will lie along that line hence moving parallel to the line $y=0.5x$.
