Question

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

Answer 1

Upon multiplication with this matrix A, it gives a vector perpendicular to the original, with a magnitude of
`√(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=√(Det|A|) (y,-x)^T`