Visualizing Linear Transformation of Unit Circle via Matrix Multiplication and Eigenvectors

Question

Consider the matrix
\begin{equation*}
A = \begin{pmatrix}
5 & 1\\
1 & 5\\
\end{pmatrix}
\end{equation*}
$\textbf{Question 1:}$ Draw the image of a unit circle after multiplying with matrix $A$

Since matrix multiplication is a linear transformation and $A$ being symmetric, I know the the circle is taken to an ellipse with origin preserved. But how do I find the major and minor axis of that ellipse? I computed the image of the unit basis vectors after applying the matrix $A$ above. Are those enough to now draw the required ellipse? Or should I pick more unit vectors at random?

$\textbf{Question 2:}$ Draw eigenvectors of $A$ in the same cartesian plane that has the unit circle and the ellipse.

I can find the eigenvectors by using $det(A-\lambda I)$ and then plotting them. However is there a way to plot/visualize them on the cartesian plane itself which contains both the unit circle and ellipse?

Answer 1

Consider the square with vertices $(\pm1,\pm1)$, which is tangent to the unit circle. The square is transformed into a rhombus with vertices $\pm(6,6)$, $\pm(4,-4)$, which is tangent to the ellipse at $\pm(5,1)$ and $\pm(1,5)$ (these are the transformed of the tangency points of circle and square).

Rhombus and ellipse are symmetric about lines $y=\pm x$, which are then the axes of symmetry of the ellipse. To find its vertices, we can exploit a nice property:

if $A$ and $B$ are two tangency points of an ellipse, $M$ the midpoint of $AB$, $O$ the center of the ellipse and $C$ the intersection point of the tangents, then points $OMC$ are aligned and the point $E$ of the ellipse between $M$ and $C$ satisfies: $OE^2=OM\cdot OC$.

(Proof: this is true for a circle, and linear transformations preserve ratios of segments on the same line).

Hence semi-axes have lengths $6$ and $4$.