Which Matrix Represents a Rotation About the Z-axis?

[unscientific]

Homework Statement

Which matrix represents a rotation?

Homework Equations

The Attempt at a Solution

It seems odd that this matrix has somewhat the form for rotation about z-axis, just that you need to swap the cos θ for the sin θ.

 

[vanhees71]

A real $3 \times 3$ matrix is a rotation iff it is an SO(3) matrix, i.e., if
$$\hat{M} \hat{M}^{\dagger}=1 \quad \text{and} \quad \mathrm{det} \; \hat{M}=1.$$

 

[D H]

It seems odd that this matrix has somewhat the form for rotation about z-axis, just that you need to swap the cos θ for the sin θ.

You didn't show very much work.

Before I help you further, did you solve the first two parts of the problem?
- Identifying which one of those two matrices represents a rotation, and
- For that matrix, finding the angle and axis of revolution.

If you didn't, do you know how to identify whether a matrix represents a rotation? Do you know how to find the eigenrotation (angle and axis of revolution) for a rotation matrix?

 

[unscientific]

You didn't show very much work.

Before I help you further, did you solve the first two parts of the problem?
- Identifying which one of those two matrices represents a rotation, and
- For that matrix, finding the angle and axis of revolution.

If you didn't, do you know how to identify whether a matrix represents a rotation? Do you know how to find the eigenrotation (angle and axis of revolution) for a rotation matrix?

The first matrix is the rotation matrix. It is a rotation matrix about z-axis.

My problem here is that the diagonals of a rotation matrix should be equal! But here clearly it's not as [1,1] = -1/2 while [2,2] = 1/2

 

[D H]

Are you sure the first matrix is a rotation matrix? Examine it carefully.

An orthogonal matrix must obey the following:
- Each row, when viewed as a vector, must be a unit vector. (The same applies for columns.)
- Two different rows, when viewed as vectors, must be orthogonal to one another. (The same applies for columns.)

In short, MM^T=I.

The determinant of an orthogonal matrix is either +1 or -1. An orthogonal matrix whose determinant is +1 represents a rotation matrix. The matrix does not represent a rotation if the determinant is -1. It represents something else. Your text or your class notes should say what.

There is an alternative to using the determinant that works only for 3x3 matrices. Does the cross product of the first two rows equal the third row or its additive inverse? You have a matrix that represents a rotation if the matrix satisfies the conditions for an orthogonal matrix and if this cross product is equal to the third row. If the conditions are met but the cross product is the additive inverse of the third row you have an improper orthogonal matrix. If the conditions are met and this cross product is neither equal to nor the inverse of the third row, that means you've made a mistake.

 

[HallsofIvy]

The first matrix is the rotation matrix. It is a rotation matrix about z-axis.

Through what angle? And what is your reasoning?

My problem here is that the diagonals of a rotation matrix should be equal! But here clearly it's not as [1,1] = -1/2 while [2,2] = 1/2

I have no idea what you mean by this. What do you mean by [1, 1] and in what sense is it equal to the number -1/2? Did you look at what the vectors (1, 0, 0) and (0, 1, 0) are mapped to?

 

[D H]

My problem here is that the diagonals of a rotation matrix should be equal! But here clearly it's not as [1,1] = -1/2 while [2,2] = 1/2

I have no idea what you mean by this. What do you mean by [1, 1] and in what sense is it equal to the number -1/2?

I think that what he means is that each of the three primitive rotations in 3 dimensional space (rotations about the x, y, and z axes) have a matrix in which one of the diagonal elements is 1 and the other two are equal to one another.

That clearly isn't the case here.

So maybe this isn't a rotation matrix. Perhaps it's that ugly beast to the right that is the rotation matrix.

 

[unscientific]

Through what angle? And what is your reasoning?


I have no idea what you mean by this. What do you mean by [1, 1] and in what sense is it equal to the number -1/2? Did you look at what the vectors (1, 0, 0) and (0, 1, 0) are mapped to?

Sorry, by [1,1] i meant the 1st row, 1st column number.

 

[unscientific]

I think that what he means is that each of the three primitive rotations in 3 dimensional space (rotations about the x, y, and z axes) have a matrix in which one of the diagonal elements is 1 and the other two are equal to one another.

That clearly isn't the case here.

So maybe this isn't a rotation matrix. Perhaps it's that ugly beast to the right that is the rotation matrix.

A^T = I.

But |A| = -1

Hence it is a rotation, followed by a reflection in the x-y plane.

Also, B^TB = I and |B| = 1

Therefore, B is the rotation matrix.

Last but not least, B is more beautiful than A.

 

[D H]

Hence it is a rotation, followed by a reflection in the x-y plane.

A rotation about what? Notice that the 3,3 element is 1. There's no reflection in the x-y plane here.

That first matrix looks like a pure reflection to me. Whether it can be composed as a reflection plus a rotation, or as a rotation plus a reflection, that's possible, but it would be pretty ugly.