Question

Show that a \(2 \times 2\) rotation matrix does not have a real eigenvalue (and, therefore, eigenvector) when the rotation angle is not an integer multiple of \(\pi\). What is the physical interpretation of this?

Short answer

A \(2 \times 2\) rotation matrix does not have a real eigenvalue when the rotation angle is not an integer multiple of \(\pi\). This means that a rotation in two dimensions doesn't have a direction that stays on the real plane (i.e., a real eigenvector) unless the rotation is by an integer multiple of \(\pi\).

Step-by-step solution

Write down the general form of rotation matrix

The rotation matrix in 2D is given by \(R(\theta) = \begin{{bmatrix}} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{{bmatrix}}\)

Set up the characteristic equation

The characteristic equation is obtained by setting the determinant of \(R - \lambda I\) to zero. Here \(I\) is the identity matrix and \(\lambda\) are the eigenvalues. The equation becomes \( \text{{det}}(R - \lambda I) = 0 \).

Calculate the determinant

Calculate the determinant of \(R - \lambda I\). It results in \( (\cos(\theta) - \lambda)^2 + \sin^2(\theta) = 0\).

Solve for eigenvalues

Solve the equation for \(\lambda\) to find the eigenvalues. We get \(\lambda = \cos(\theta) \pm i \sin(\theta)\). These are complex numbers (except when \(\theta\) is an integer multiple of \(\pi\), in which case sin and cos are 0 or 1, giving real eigenvalues).

Interpret the result

The physical interpretation of this result is that a rotation in two dimensions doesn't have a direction that stays on the real plane (i.e., a real eigenvector) unless the rotation is by an integer multiple of \(\pi\), which means a complete turnaround. A non-zero rotation takes every vector off the real plane, therefore it only has complex eigenvalues (and associated eigenvectors).

Key concepts

Understanding 2D Transformations

2D transformations involve changing the position or orientation of objects in a two-dimensional plane. Common transformations include translation, scaling, and rotation.
Rotation is a specific transformation where a shape or point is turned around a fixed point or axis. In 2D space, this is often represented using a rotation matrix.
The rotation matrix allows us to calculate the new position of a point after it has been rotated by a specified angle, \(\theta\). Specifically, the rotation matrix \(R(\theta)\) is:

• \(\begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix}\)

By applying this matrix to the coordinates of a point, you can determine where the point will be after the rotation.
Understanding this concept is essential in fields like computer graphics, physics, and robotics, where objects need to be manipulated frequently.

Exploring Eigenvalues

Eigenvalues are a crucial concept in linear algebra and are associated with square matrices. They provide insights into the properties of a matrix and the transformations it represents.

To find eigenvalues, we solve the equation \(\text{det}(A - \lambda I) = 0\), where \(A\) is our matrix, \(\lambda\) represents the eigenvalues, and \(I\) is the identity matrix. The solution tells us how much the eigenvectors, associated with these eigenvalues, will be stretched or compressed during the transformation.

In the case of a rotation matrix, the characteristic equation becomes \((\cos(\theta) - \lambda)^2 + \sin^2(\theta) = 0\). Solving this gives us the eigenvalues \(\lambda = \cos(\theta) \pm i \sin(\theta)\), indicating that they are complex numbers, thus reflecting the nature of rotational transformations.

This outcome demonstrates an essential property: real eigenvalues only emerge when \(\theta\) is an integer multiple of \(\pi\), meaning only then do rotations align with real directions.

The Importance of Eigenvectors

Eigenvectors are vectors that correspond to eigenvalues in matrix transformations. For any given matrix \(A\) and an eigenvalue \(\lambda\), the eigenvector \(\mathbf{v}\) satisfies \(A \mathbf{v} = \lambda \mathbf{v}\). This relationship signifies that the transformation represented by \(A\) stretches or compresses \(\mathbf{v}\) by the factor \(\lambda\).

In a 2D rotation, however, it becomes evident that when the angle \(\theta\) is not a multiple of \(\pi\), the concept of real eigenvectors quickly fades. This is because the results are complex, meaning no real-line direction remains unchanged after the rotation. This complexity signifies that the vectors switch into a plane that is imaginative (not physically possible in simple real-world geometry).

The takeaway is that for practical, real eigenvectors to exist in 2D rotations, the situation must align perfectly—like when a complete rotation by multiples of \(\pi\) occurs. In computational fields, understanding the role and nature of eigenvectors is vital for interpreting matrix transformations effectively.