Can you have complex eigenvalues

So, now that we have the eigenvalues recall that we only need to get the eigenvector for one of the eigenvalues since we can get the second eigenvector for free from the first eigenvector.

Also try to clear out any fractions by appropriately picking the constant. This will make our life easier down the road. So, as we can see there are complex numbers in both the exponential and vector that we will need to get rid of in order to use this as a solution. Doing this gives us,. This means that we can use them to form a general solution and they are both real solutions.

The only thing that we really need to concern ourselves with here are whether they are rotating in a clockwise or counterclockwise direction. This is easy enough to do. The only way that this can be is if the trajectories are traveling in a clockwise direction.

Note in this last example that the equilibrium solution is stable and not asymptotically stable. In this example the trajectories are simply revolving around the equilibrium solution and not moving in towards it.

So the eigenvectors of the above matrix A associated to the eigenvalue i are given by. Let us summarize what we did in the above example. Summary : Let A be a square matrix. Assume is a complex eigenvalue of A. In order to find the associated eigenvectors, we do the following steps: 1. Write down the associated linear system. In general, it is normal to expect that a square matrix with real entries may still have complex eigenvalues. One may wonder if there exists a class of matrices with only real eigenvalues.

See this important note in Section 5. One should regard the rotation-scaling theorem as a close analogue of the diagonalization theorem in Section 5. Before continuing, we restate the theorem as a recipe:.

We saw in the above examples that the rotation-scaling theorem can be applied in two different ways to any given matrix: one has to choose one of the two conjugate eigenvalues to work with.

The matrices B and B A are similar to each other. The discussion that follows is closely analogous to the exposition in this subsection in Section 5. There are three different cases. For example,. There are four cases:. It says essentially that a matrix is similar to a matrix with parts that look like a diagonal matrix, and parts that look like a rotation-scaling matrix.

Suppose that for each real or complex eigenvalue, the algebraic multiplicity equals the geometric multiplicity. The block diagonalization theorem is proved in the same way as the diagonalization theorem in Section 5. Since there are three distinct eigenvalues, they have algebraic and geometric multiplicity one, so the block diagonalization theorem applies to A. Objectives Learn to find complex eigenvalues and eigenvectors of a matrix. Learn to recognize a rotation-scaling matrix, and compute by how much the matrix rotates and scales.

Pictures: the geometry of matrices with a complex eigenvalue.