# 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.).

-

## 1 Answer

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 $\ker(A-\lambda I)$.

Nothing is changing in the real or complex case. The only problem may come in when the matrices get large and you have to factor large polynomials (which may not be possible in closed form). But this difficulty holds in the real case, too.

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 $0,1+2i,1-2i$.

-