Proving a certain orthogonal matrix is a rotation matrix

In summary, the matrix ##U## is a rotation matrix that corresponds to either ##x=\cos\theta## or ##x=\sin\theta##.
  • #1
ELB27
117
15

Homework Statement


Let ##U## be a ##2\times 2## orthogonal matrix with ##\det U = 1##. Prove that ##U## is a rotation matrix.

Homework Equations

The Attempt at a Solution


Well, my strategy was to simply write the matrix as
$$U = \begin{pmatrix}
a & b\\
c & d
\end{pmatrix}$$
and using the given properties to solve for ##a,b,c## and ##d##. I have 4 equations:
1. Determinant = 1
2. ##U^TU = I## where ##I## is identity
where the last property gives me 3 equations (one of the entries is redundant). Thus, I have 4 equations in 4 unknowns. When I solve them, I get 1 free variable, and my matrix turns out to be of the form:
$$U = \begin{pmatrix}
x & \mp \sqrt{1-x^2}\\
\pm \sqrt{1-x^2} & x
\end{pmatrix}$$
and since all entries must be real (orthogonal matrix) we have the constraint on ##x##:
$$-1≤x≤1$$
Clearly, these properties are satisfied if we let ##x=\cos\theta## or ##x=\sin\theta##, thus obtaining a rotation matrix in its standard notation. However, I do not see how to prove that these trigonometric functions are the only possible solutions. Also, how does one define formally and rigorously a rotation matrix? Only as a matrix of cosines and sines (with the appropriate values of determinant etc.)?

Any suggestions/comments will be greatly appreciated!
 
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  • #2
ELB27 said:
Clearly, these properties are satisfied if we let x=cosθ or x=sinθ, thus obtaining a rotation matrix in its standard notation. However, I do not see how to prove that these trigonometric functions are the only possible solutions. Also, how does one define formally and rigorously a rotation matrix? Only as a matrix of cosines and sines (with the appropriate values of determinant etc.)?
There is no need for that. You've done all things needed. You have shown that for all allowed values of x, there exists a ##\theta## that either ##x=\sin\theta ## or ##x=\cos\theta##. It means for all allowed values of x, this matrix corresponds to a rotation. It may correspond to many other things, but that doesn't matter. All we care about now, is that it corresponds to a rotation. So you're done with this question.
 
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  • #3
Shyan said:
There is no need for that. You've done all things needed. You have shown that for all allowed values of x, there exists a ##\theta## that either ##x=\sin\theta ## or ##x=\cos\theta##. It means for all allowed values of x, this matrix corresponds to a rotation. It may correspond to many other things, but that doesn't matter. All we care about now, is that it corresponds to a rotation. So you're done with this question.
Ah, I think I get it. Basically, the question can be rephrased as "Prove the ##U## can be represented as a rotation matrix."? Thus my proof will end as:
##x=\cos\theta \ ∀x## for some angle ##\theta## and thus, ##U## is always a rotation matrix about some angle ##\theta##.

Thanks for the reply!
 

1. What is an orthogonal matrix?

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, meaning they are mutually perpendicular and have a length of 1. This means that when multiplied with its transpose, the result is the identity matrix.

2. What is a rotation matrix?

A rotation matrix is a specialized type of orthogonal matrix that represents a rotation in n-dimensional space. It is used to rotate a vector or set of coordinates by a certain angle around a fixed point.

3. How do you prove that a matrix is orthogonal?

To prove that a matrix is orthogonal, you can use the property that when multiplied with its transpose, the result should be the identity matrix. This can be done by multiplying the matrix with its transpose and checking if the resulting matrix is the identity matrix. If it is, then the matrix is orthogonal.

4. How do you prove that a matrix is a rotation matrix?

To prove that a matrix is a rotation matrix, you need to show that it is an orthogonal matrix and that it has a determinant of 1. This shows that the matrix has no scaling or reflection components, and only represents a rotation in n-dimensional space.

5. What are some real-life applications of rotation matrices?

Rotation matrices have various applications in fields such as computer graphics, robotics, and physics. They are used to rotate objects in 3D computer graphics, orient cameras in robotics, and represent the motion of rigid bodies in physics simulations.

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