How to compute eigennumbers of matrix 2x2 A: $$ A = \left[\begin{array}{cc} 3i&1\\ -1&3i \end{array}\right]$$
I got the polynomial I think which is $a^2-6ia-8$ but am not sure what the eigenvalues would be.
The full problem is finding exp of $\frac \pi 6 A$. Please help.
As you said $p_A(x)=x^2-6ix-8$ so eigenvalues are $x$ such that $p_A(x)=x^2-6ix-8=0$ and so you get $x_1=2i$ $x_2=4i$. To find $e^{\pi A/6}$ i suggest you to consider if this matrix is diagonalizable, and if yes find the matrix $P$ such that $P^{-1}AP$ is diagonal.