When Are Eigenvalues Complex: Key Concepts and Applications

Understanding when eigenvalues are complex is essential in various fields of science and engineering, particularly in linear algebra and the analysis of dynamical systems. Eigenvalues can be complex under specific conditions, such as dealing with non-symmetric matrices or matrices with complex entries. This article explores these conditions and their implications in different contexts.

Conditions for Complex Eigenvalues

There are several scenarios where eigenvalues can be complex. These include the use of non-symmetric matrices, complex matrices, the characteristic polynomial, and stability analysis. Below, we delve into each of these scenarios and provide examples to illustrate the concepts.

Non-Symmetric Matrices

For real-valued matrices that are not symmetric, the eigenvalues can be complex. This is because the eigenvalues of a real n times n matrix can be complex numbers, and they occur in conjugate pairs when the matrix has real entries. If the matrix is not symmetric, its eigenvalues are not necessarily real. For example, consider the following matrix:

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

The characteristic polynomial of matrix A is given by:

det(A - lambda I) detbegin{pmatrix} -lambda -1 1 -lambda end{pmatrix} lambda^2 1

The roots of this polynomial are lambda i and lambda -i, which are complex eigenvalues.

Complex Matrices

If a matrix has complex entries, its eigenvalues can also be complex. This is true regardless of whether the matrix is Hermitian or not. For instance, consider a matrix with complex entries:

B begin{pmatrix} 2 3i -3i 2 end{pmatrix}

The characteristic polynomial of matrix B is:

det(B - lambda I) detbegin{pmatrix} 2 - lambda 3i -3i 2 - lambda end{pmatrix} (2 - lambda)^2 9 lambda^2 - 4lambda 13

The roots of this polynomial are lambda 2 pm 3i, which are complex eigenvalues.

Characteristic Polynomial

The eigenvalues of a matrix are the roots of its characteristic polynomial. If the characteristic polynomial has no real roots, such as a quadratic polynomial with a negative discriminant, the eigenvalues will be complex. For example, consider the characteristic polynomial:

lambda^2 1 0

The roots of this polynomial are lambda i and lambda -i, which are complex eigenvalues.

Stability Analysis

In the context of differential equations and dynamical systems, complex eigenvalues can indicate oscillatory behavior. For instance, in a system described by a matrix with complex eigenvalues, the system may exhibit spiraling behavior in its phase portrait. This is relevant in various fields, including mechanical and electrical engineering.

Consider a matrix that describes a linear system:

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

The eigenvalues of matrix C are lambda i and lambda -i, which indicate oscillatory behavior in the system. This can be visualized in a phase portrait as a spiral.

Rotation and Complex Eigenvalues

Another situation where complex eigenvalues arise is when there is a rotation involved in the transformation. For example, a rotation by an angle theta in the plane mathbb{R}^2 is given by the matrix:

A begin{bmatrix} costheta -sintheta sintheta costheta end{bmatrix}

The eigenvalues of this matrix are lambda e^{pm itheta} costheta pm isintheta. If you compose a rotation with most other linear transformations, such as expansions, contractions, stretches along a vector, and shears, the resulting transformation will have a rotary aspect and complex eigenvalues.

For instance, consider a transformation matrix D that is a combination of a rotation and a contraction:

D begin{bmatrix} costheta -sintheta sintheta costheta end{bmatrix} cdot begin{bmatrix} 2 0 0 1/2 end{bmatrix}

The eigenvalues of matrix D are still complex and given by lambda 2e^{pm itheta} and lambda frac{1}{2}e^{pm itheta}. This shows that the rotary component persists in the combined transformation.

Conclusion

In summary, eigenvalues are complex primarily when dealing with non-symmetric matrices, matrices with complex entries, or when the characteristic polynomial has no real solutions. Understanding these conditions is crucial for analyzing the behavior of dynamical systems and predicting various physical phenomena. Whether it's oscillatory behavior in mechanical systems or the rotary aspect in combined transformations, complex eigenvalues play a vital role in these analyses.