Rotation matrix vs regular matrix

In summary, when calculating eigenvalues and eigenvectors for rotation matrices, the existence of real eigenvalues will depend on the angle of rotation. Most angles will result in complex eigenvalues. However, the determinant of a rotation matrix is always 1, which means there will always be eigenvalues and eigenvectors that can be calculated. The characteristic polynomial for a rotation matrix is x^2 - (2cosθ)x + 1, which has real solutions only when cosθ = ±1. For angles that are not integer multiples of pi, this will result in complex eigenvalues. This fact is related to the concept of rotation.
  • #1
dmoney123
32
1
Can you calculate eigenvalues and eigenvectors for rotation matrices the same way you would for a regular matrix?

If not, what has to be done differently?
 
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  • #2
it don't see why not. the existence of real eigenvalues will depend on the angle of rotation (most angles will give complex eigenvalues).
 
  • #3
The determinant of a rotation matrix should always be 1 (since it preserves length) so there should always be eigenvalues and eigenvectors that can be calculated given a rotation matrix.
 
  • #4
chiro said:
The determinant of a rotation matrix should always be 1 (since it preserves length) so there should always be eigenvalues and eigenvectors that can be calculated given a rotation matrix.

the characteristic polynomial of:

[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]

is:

[tex]x^2 - (2\cos\theta)x + 1[/tex]

which has real solutions only when:

[tex]4\cos^2\theta - 4 \geq 0 \implies \cos\theta = \pm 1[/tex]

for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.
 
  • #5
Deveno said:
the characteristic polynomial of:

[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]

is:

[tex]x^2 - (2\cos\theta)x + 1[/tex]

... and the eigenvalues are [itex]\cos\theta \pm i \sin\theta[/itex]. Now, I wonder what that fact might have to do with "rotation"... :smile:
 
  • #6
Deveno said:
the characteristic polynomial of:

[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]

is:

[tex]x^2 - (2\cos\theta)x + 1[/tex]

which has real solutions only when:

[tex]4\cos^2\theta - 4 \geq 0 \implies \cos\theta = \pm 1[/tex]

for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.

What has that got to do with what I said?
 

1. What is the difference between a rotation matrix and a regular matrix?

A rotation matrix is a specialized type of matrix that is used to represent rotations in space. It is a square matrix with a special structure that allows for easy representation of rotation transformations. On the other hand, a regular matrix is a generic matrix that can represent any linear transformation, not just rotations.

2. How is a rotation matrix different from a regular matrix in terms of dimensions?

A rotation matrix is always a square matrix, meaning it has an equal number of rows and columns. This is because rotations preserve the dimensions of objects in space. In contrast, a regular matrix can have any number of rows and columns, depending on the transformation it represents.

3. Can a regular matrix be used to represent a rotation?

Yes, a regular matrix can be used to represent a rotation. However, it would require a larger matrix with more parameters compared to a rotation matrix, which has only 9 parameters. This is because a regular matrix has to account for all possible linear transformations, while a rotation matrix is specifically designed for rotations.

4. What are the advantages of using a rotation matrix over a regular matrix?

The main advantage of using a rotation matrix is that it simplifies the representation of rotations in space. It is also more efficient computationally since it has fewer parameters compared to a regular matrix. Additionally, rotation matrices have special properties that make them useful for other applications, such as 3D graphics and robotics.

5. Can a regular matrix be converted into a rotation matrix?

Yes, it is possible to convert a regular matrix into a rotation matrix, but it may not always be a straightforward process. Depending on the specific transformation represented by the regular matrix, it may require some mathematical operations, such as finding eigenvalues and eigenvectors, to convert it into a rotation matrix.

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