2x2 eigenvectors as you drag the matrix

What you are seeing: a 2x2 matrix $M = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$ acting on the unit circle. The blue ellipse is the image of the unit circle under $M$ (semi-axes equal to the singular values). The accent arrows are the eigenvectors of $M$, scaled by their eigenvalues. The eigenvalue formula $\lambda_\pm = (a+d)/2 \pm \sqrt{(a+d)^2/4 - (ad - bc)}$ has real solutions when the discriminant is non-negative.

Drag the sliders for $a, b, c, d$. With $a = d$ and $b = -c$ you get a rotation; the eigenvalues become complex and the eigenvector arrows disappear (the readout flags the complex spectrum). With $b = c$ (symmetric $M$) the eigenvectors are exactly orthogonal. With $b = c = 0$ the eigenvectors stay axis-aligned regardless of $a, d$.