Can a Real Matrix Have Complex Eigenvalues?

When we first approach the concept of eigenvalues, many might assume that real matrices can only have real eigenvalues. However, this is a misconception, as real matrices can indeed have complex eigenvalues under certain conditions. Let's dive deeper into this fascinating topic and explore the conditions under which real matrices can have complex eigenvalues.

Real Matrices and Eigenvalues

For a real matrix to have real eigenvalues, one of the most commonly known conditions is that the matrix must be symmetric. Symmetry in a matrix, denoted by (A A^T), ensures that all its eigenvalues are real. However, a real matrix does not necessarily have to be symmetric to have complex eigenvalues.

Antisymmetric Matrices and Complex Eigenvalues

Antisymmetric matrices, defined as (A -A^T), are a special type of real matrix that always have complex eigenvalues. An example of an antisymmetric matrix is the 2x2 matrix:

$$begin{pmatrix} 0 -1 1 0 end{pmatrix}$$

This matrix has the eigenvalues ( pm i ), which are purely imaginary. This is a key characteristic of antisymmetric matrices, as they always have eigenvalues that are purely imaginary.

General Case: Any Real Matrix with Complex Eigenvalues

While symmetric matrices and antisymmetric matrices provide clear examples of conditions under which real matrices can have complex eigenvalues, the possibility of complex eigenvalues is not limited to just these types of matrices. In fact, any real matrix can potentially have complex eigenvalues, provided that the characteristic polynomial of the matrix includes complex roots.

Example of a Matrix with Complex Eigenvalues

Consider a 2x2 matrix with the characteristic polynomial (x^2 - 3x 1 0). The eigenvalues of this matrix can be determined by solving the quadratic equation:

$$x^2 - 3x 1 0$$

Using the quadratic formula, (x frac{-b pm sqrt{b^2 - 4ac}}{2a}), we find the eigenvalues:

$$x frac{3 pm sqrt{3^2 - 4(1)(1)}}{2(1)} frac{3 pm sqrt{9 - 4}}{2} frac{3 pm sqrt{5}}{2}$$

Although this quadratic equation does not yield complex eigenvalues, consider a different example:

For the characteristic polynomial (x^2 - 3x 3i 0), the eigenvalues would be:

$$x frac{3 pm sqrt{3^2 - 4(1)(3i)}}{2(1)} frac{3 pm sqrt{9 - 12i}}{2} frac{3 pm (3i - 1)}{2} {3 (3i - 1), 3 - (3i - 1)} {3 3i - 1, 3 - 3i 1} {2 3i, 4 - 3i}$$

Here, we see that the eigenvalues are complex.

Conclusion

In summary, while symmetric matrices have the property that all their eigenvalues are real, real matrices in general can have complex eigenvalues. Antisymmetric matrices are a special case where all eigenvalues are purely imaginary. The key is to understand that the eigenvalues of a real matrix can be complex, provided the characteristic polynomial has complex roots. Understanding these concepts can provide deeper insights into linear algebra and the behavior of real matrices.

References

1. MathWorld - Antisymmetric Matrix

2. Wikipedia - Symmetric Matrix