Question

If a 2x2 matrix has complex eigenvalues, what does this matrix do to vectors upon multiplication?

Answer 1

Answer:

Upon multiplication with this matrix A, it gives a vector perpendicular to the original, with a magnitude of $\sqrt{Det|A|}$ times of the original.

If a 2x2 matrix  A has complex eigenvalues, this is an antisymmetric matrix.

Therefore:

$A=\left[\begin{array}{cc}0&a\\-a&0\end{array}\right] \\Det|A|=a^2$

if we multiply (x,y) to this matrix A:

$A\cdot (x,y)^T=\left[\begin{array}{cc}0&a\\-a&0\end{array}\right](x,y)^T=(ay,-ax)^T=\sqrt{Det|A|} (y,-x)^T$