Determine the Jordan canonical form of the matrix [[5, 4], [-1, -2]].
We first find the eigenvalues of the matrix. The characteristic polynomial is given by:

\[(5 - \lambda)(-2 - \lambda) - 4(-1) = \lambda^2 + 3\lambda + 6.\]

The discriminant of this quadratic is $3^2 - 4(6) = -15$, which is negative, so there are no real eigenvalues. Therefore, the Jordan canonical form must be a block diagonal matrix with two complex conjugate eigenvalues.

Let $\lambda_1 = -\frac{3}{2} + \frac{\sqrt{15}}{2}i$ and $\lambda_2 = -\frac{3}{2} - \frac{\sqrt{15}}{2}i$. Then we can write the matrix as

\[A = PJP^{-1},\]

where

\[J = \begin{pmatrix} -\frac{3}{2} + \frac{\sqrt{15}}{2}i & 0 \\ 0 & -\frac{3}{2} - \frac{\sqrt{15}}{2}i \end{pmatrix}\]

and $P$ is some invertible matrix. Thus, the Jordan canonical form of the matrix is

\[\boxed{\begin{pmatrix} -\frac{3}{2} + \frac{\sqrt{15}}{2}i & 0 \\ 0 & -\frac{3}{2} - \frac{\sqrt{15}}{2}i \end{pmatrix}}.\]