Finding the transformation of a matrix

  • Thread starter Redwaves
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  • #1
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Homework Statement:
Finding the transformation of a matrix
Relevant Equations:
##\begin{bmatrix}
cos \theta & sin \theta \\
sin \theta & -cos \theta
\end{bmatrix}##
I have the matrix above and I have to find which transformation is that.
##\begin{bmatrix}
cos \theta & sin \theta \\
sin \theta & -cos \theta
\end{bmatrix}##

For a vector ##\vec{v}##

TPa8NJS.png


##v_x' = v_x cos \theta + v_y sin \theta##
##v_y' = v_x sin \theta - v_y cos \theta##

If ##\phi## is the angle between x axis and the vector ##\vec{v}##, then ##v_x = r cos \theta ## and ##v_y = r sin \theta##

Thus,
##v_x = r cos \phi cos \theta + r sin\phi sin\theta## = ##r cos(\phi - \theta)##
##v_y = r cos \phi sin \theta - r sin\phi cos\theta## = - ##r sin(\phi - \theta)##

From that, the transformation seems to be an rotation of ##\phi - \theta## clockwise and then a reflection over the x axis. Is this correct?
 
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Answers and Replies

  • #2
PeroK
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Where did ##\phi## come from?
 
  • #3
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Where did ##\phi## come from?
Between x and ##\vec{v}##
 
  • #4
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Between x and ##\vec{v}##
What has that to do with a matrix that is a function of ##\theta##?
 
  • #5
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What has that to do with a matrix that is a function of ##\theta##?
##\vec{v}## is an arbitrary vector to help me figure out what the matrix does.
 
  • #6
PeroK
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##\vec{v}## is an arbitrary vector to help me figure out what the matrix does.
I understand that, but what the matrix does can only depend on ##\theta## and not on some other angle. I'm struggling to see how another angle could be involved in the general matrix operation.
 
  • #7
PeroK
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PS if you know how to write the rotation and reflection matrices, then you can check your answer. Do you know how?
 
  • #8
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PS if you know how to write the rotation and reflection matrices, then you can check your answer. Do you know how?
I watched youtube videos and they use the same way as I did to find the rotation matrix. A vector ##\vec{v}## and an angle ##\phi## between x and ##\vec{v}##. Otherwise, I don't know.
 
  • #9
PeroK
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I watched youtube videos and they use the same way as I did to find the rotation matrix. A vector ##\vec{v}## and an angle ##\phi## between x and ##\vec{v}##. Otherwise, I don't know.
Okay, let's check whether the matrix is a rotation by ##\phi - \theta## clockwise followed by a reflection in the ##x## axis. How would you check that?

Hint: you could do an Internet search for "rotation matrix" and "reflection matrix".
 
  • #10
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Hint: you could do an Internet search for "rotation matrix" and "reflection matrix".
I did, but this matrix is neither a rotation or a reflection matrix.
 
  • #11
PeroK
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I did, but this matrix is neither a rotation or a reflection matrix.
I know! Maybe you need to think about matrix operations?
 
  • #12
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PS why don't we use the vector ##(1, 2)## as an example? Calculate what your matrix does to that.

And, we could set ##\theta = \frac{\pi}{4}## perhaps. Just to check what happens to a given vector for a simple value of ##\theta##.

That should let you check your answer.
 
  • #13
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PS why don't we use the vector ##(1, 2)## as an example? Calculate what your matrix does to that.

And, we could set ##\theta = \frac{\pi}{4}## perhaps. Just to check what happens to a given vector for a simple value of ##\theta##.

That should let you check your answer.
##x' = \frac{3\sqrt{2}}{2}##
##y' = -\frac{\sqrt{2}}{2}##
It seems to be the same transformation as I said, no?
 
  • #14
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##x' = \frac{3\sqrt{2}}{2}##
##y' = -\frac{\sqrt{2}}{2}##
It seems to be the same transformation as I said, no?
Where's ##\phi##?
 
  • #15
PeroK
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##x' = \frac{3\sqrt{2}}{2}##
##y' = -\frac{\sqrt{2}}{2}##
It seems to be the same transformation as I said, no?
PS No. It's not a rotation followed by a reflection.
 
  • #16
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I mean, if you draw the new vector using both way the new vector is in the same position.
 
  • #17
PeroK
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I mean, if you draw the new vector using both way the new vector is in the same position.
That's not what I get. What about looking at the basis vectors ##(1,0)## and ##(0,1)##?

When you say "a clockwise rotation of ##\phi - \theta##", what does that mean?
 
  • #18
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When you say "a clockwise rotation of ##\phi - \theta##", what does that mean?
##\theta## is the angle between ##\vec{v}'## and the x axis.
Thus. ##\phi - \theta## is the angle between ##\vec{v}## and ##\vec{v}'##.
##\vec{v}'## is closer to the x axis.

Is it clear?
 
  • #19
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##\theta## is the angle between ##\vec{v}'## and the x axis.
Thus. ##\phi - \theta## is the angle between ##\vec{v}## and ##\vec{v}'##.
##\vec{v}'## is closer to the x axis.

Is it clear?
This is not right. The matrix you have is a reflection in the ##x## axis followed by a rotation by ##\theta##. By convention rotations are anti-clockwise.

Alternatively, it's a rotation of ##-\theta## followed by reflection in the ##x## axis.

There's no ##\phi## involved, regardless of what you've seen on Youtube.

The other thing you are missing is that you can use matrix multiplication to generate a rotation followed by a reflection or vice versa.

That's how you could have checked your answer.
 
  • #20
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In this case how can I find the rotation matrix and the reflection? I was able to find both matrices using my way. Otherwise, I don't see how.

What's ##\theta##? the angle between x and v ?
 
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  • #21
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In this case how can I find the rotation matrix and the reflection? I was able to find both matrices using my way. Otherwise, I don't see how.
The first thing I would do is check what it does to the basis vectors:

##(1, 0) \rightarrow (\cos \theta, \sin \theta)##

##(0, 1) \rightarrow (\sin \theta, -\cos \theta)##

You can see from that that you have to do the reflection in the ##x## axis first.

Then I would check that ##R_{\theta} R_x## equals the matrix you were given. Note the order in the matrix multiplication. This is because you want the reflection first:$$R_{\theta} R_x v$$
 
  • #22
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What's ##R## ?
How can I prove the reflection and the rotation without using any specific vector.

Is ##\theta## the angle between x and v ?

How can I prove the rotation matrix without using ##\phi##, that is what I used.

Everything seems less clear.
 
  • #23
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I think what you say is exactly what I say in a different way and that's what confuse me.

##v_x = r cos (\phi - \theta)##
##v_y = - r sin (\phi - \theta)##
is a rotation of ##-\theta## from the initial angle like you say and a reflection in the x axis.
 
  • #24
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What's ##R## ?
How can I prove the reflection and the rotation without using any specific vector.

Is ##\theta## the angle between x and v ?

How can I prove the rotation matrix without using ##\phi##, that is what I used.

Everything seems less clear.
##R_{\theta}## is the matrix representing a (counterclockwise) rotation by ##\theta##. ##R_x## is the matrix representing reflection in the ##x## axis.

You can prove it by matrix multiplication. Once you have the matrix equation, there is no need to consider vectors, either specific or general.

There's no need to involve ##\phi##. The video you saw has confused you. It's possible to represent a vector using a polar angle ##\phi##, but that is ultimately irrelevant to the matrix itself.
 
  • #25
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You can prove it by matrix multiplication. Once you have the matrix equation, there is no need to consider vectors, either specific or general.
This is a lot of trial and error, no?
Assuming I know the transformations matrix and I don't have to prove them before using them.
 

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