# How do you find the (complex) eigenvalue and each eigenspace over C

My book only has eigenvalue and eigenspace and does not say anything about complex eigenvalue and eigenspace.

$$A = \begin{pmatrix} 0&4\\-1 & 0 \end{pmatrix}\hspace{10pt}B =\begin{pmatrix} -1&2\\-1 & 3 \end{pmatrix}\hspace{10pt}C = \begin{pmatrix} 2&2&-1\\-4 & 1&2 \\2 & 2 & -1\end{pmatrix}$$

(for the $2\times2$ matrices, use the quadratic formula to find the complex eigenvalues; for the $3\times3$ matrix, first find an integer eigenvalue, call it $1$, by the usual method (i.e. checking divisors of the constant term of the characteristic polynomial), then divide $1$ into the characteristic polynomial to get a quadratic polynomial and find the remaining two complex eigenvalues by the quadratic formula.).

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In the $2\times 2$ or $3\times 3$ case you calculate the eigenvalues and the eigenspace in the same way: Find the roots of $\det(A−λI)$ to find the eigenvalues. To find the eigenspace you need to calculate $\ker(A-\lambda I)$.
For instance, you can calculate that $$\det(C-\lambda I)=-\lambda^3+2\lambda^2-5\lambda=\lambda(\lambda-(1+2i))(\lambda-(1-2i)).$$So the eigenvalues are $0,1+2i,1-2i$.