Proving the Existence of a Rotation Matrix from Given Relations

In summary, the angle θ is such thatA = ##\left( \begin{array}{ccc}cos(\theta) & sin(\theta)\\sin(\theta) & -cos(\theta)\\\end{array} \right) ##
  • #1
whatisreality
290
1

Homework Statement


Let A∈M2x2(ℝ) such that ATA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that

A = ##
\left( \begin{array}{cc}
cos(\theta) & sin(\theta)\\
sin(\theta) & -cos(\theta)\\
\end{array} \right) ##

It is not sufficient to show that this matrix satisfies the specified relations.

Homework Equations

The Attempt at a Solution


Where do I start with this?! I'm supposed to get from ATA = I to the rotation matrix! I have looked at several proofs online. But they seem to either use eigenvalues and vectors (which we haven't done, so can't use them!) or don't mention the properties I've been given.

I know a few things that might be useful.
##(A^T)^{-1} = (A^{-1})^T##
And ##det(A^T) = det(A)##
Also, I know that the matrix A is orthogonal.

Don't know how to start!
 
Physics news on Phys.org
  • #2
whatisreality said:

Homework Statement


Let A∈M2x2(ℝ) such that ATA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that

A = ##
\left( \begin{array}{cc}
cos(\theta) & sin(\theta)\\
sin(\theta) & -cos(\theta)\\
\end{array} \right) ##

It is not sufficient to show that this matrix satisfies the specified relations.

Homework Equations

The Attempt at a Solution


Where do I start with this?! I'm supposed to get from ATA = I to the rotation matrix! I have looked at several proofs online. But they seem to either use eigenvalues and vectors (which we haven't done, so can't use them!) or don't mention the properties I've been given.

I know a few things that might be useful.
##(A^T)^{-1} = (A^{-1})^T##
And ##det(A^T) = det(A)##
Also, I know that the matrix A is orthogonal.

Don't know how to start!

Start with
[tex] A = \pmatrix{a &b \\ c& d} [/tex]

Evaluate ##P_1 = A A^T## and ##P_2 = A^T A##. You need both ##P_1 = I## and ##P_2 =I##, and those will give you several equations that the entries ##a,b,c,d## must satisfy. You also need ##\det(A) = 1##, giving you ##ad - bc = 1##.
 
  • Like
Likes whatisreality
  • #3
If you have ##a^2 + b^2 = 1##, can you show that there exists ##\theta## such that ##a = cos\theta## and ##b = sin\theta##?
 
  • #4
Ray Vickson said:
Start with
[tex] A = \pmatrix{a &b \\ c& d} [/tex]

Evaluate ##P_1 = A A^T## and ##P_2 = A^T A##. You need both ##P_1 = I## and ##P_2 =I##, and those will give you several equations that the entries ##a,b,c,d## must satisfy. You also need ##\det(A) = 1##, giving you ##ad - bc = 1##.
Got there. Thank you!
 

Related to Proving the Existence of a Rotation Matrix from Given Relations

1. What is a rotation matrix?

A rotation matrix is a 3x3 matrix that is used to describe a rotation in 3-dimensional space. It is typically represented by the symbol R and is used to perform rotations in computer graphics, robotics, and other fields of science and engineering.

2. How is a rotation matrix calculated?

A rotation matrix is calculated using trigonometric functions and a set of rotation angles. The specific calculations depend on the type of rotation (e.g. rotation around the x-axis, y-axis, or z-axis) and the direction of rotation (clockwise or counterclockwise).

3. What is the proof for the formula of a rotation matrix?

The proof for the formula of a rotation matrix involves using basic trigonometric identities and properties of matrices. It can be derived from the geometric definition of a rotation in 3-dimensional space and the properties of vector transformations.

4. Why is a rotation matrix useful?

A rotation matrix is useful because it allows for precise and efficient calculations of rotations in 3-dimensional space. It is also used in computer graphics to rotate objects and in robotics to control the movement of robotic arms and joints.

5. Can a rotation matrix be used for 2-dimensional rotations?

Yes, a rotation matrix can be used for 2-dimensional rotations by simply ignoring the third dimension and performing the rotation in a 2-dimensional space. In this case, the rotation matrix becomes a 2x2 matrix instead of a 3x3 matrix.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
9
Views
2K
  • Precalculus Mathematics Homework Help
Replies
24
Views
2K
  • Precalculus Mathematics Homework Help
2
Replies
69
Views
3K
Replies
2
Views
1K
  • Mechanical Engineering
Replies
3
Views
552
  • Precalculus Mathematics Homework Help
2
Replies
54
Views
2K
  • Precalculus Mathematics Homework Help
Replies
16
Views
4K
  • Precalculus Mathematics Homework Help
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
10
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
1K
Back
Top