Finding the transformation of a matrix

In summary: 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.
  • #1
Redwaves
134
7
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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  • #2
Where did ##\phi## come from?
 
  • #3
PeroK said:
Where did ##\phi## come from?
Between x and ##\vec{v}##
 
  • #4
Redwaves said:
Between x and ##\vec{v}##
What has that to do with a matrix that is a function of ##\theta##?
 
  • #5
PeroK said:
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
Redwaves said:
##\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
PS if you know how to write the rotation and reflection matrices, then you can check your answer. Do you know how?
 
  • #8
PeroK said:
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.
 
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  • #9
Redwaves said:
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
PeroK said:
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
Redwaves said:
I did, but this matrix is neither a rotation or a reflection matrix.
I know! Maybe you need to think about matrix operations?
 
  • #12
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
PeroK said:
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
Redwaves said:
##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
Redwaves said:
##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
I mean, if you draw the new vector using both way the new vector is in the same position.
 
  • #17
Redwaves said:
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
PeroK said:
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
Redwaves said:
##\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
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
Redwaves said:
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
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
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
Redwaves said:
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
PeroK said:
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.
 
  • #26
Redwaves said:
This is a lot of trial and error, no?
Assuming I know the transformations matrix.
No, because by checking the action on the basis vectors first you should be able to see what the matrix does. Checking the matrix multiplication just confirms that.
 
  • #28
Redwaves said:
Is the reflection matrix
##\begin{bmatrix}
cos 2\theta & sin 2\theta \\
sin 2 \theta & - cos 2\theta
\end{bmatrix}##

Because I don't get the starting matrix.
The reflection matrix is:
$$\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}$$
You can check that matrix maps ##(x, y)## to ##(x, -y)##

There's no ##\theta## in a reflection in the ##x## axis.
 
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  • #29
I see,
so it's a reflection followed by a rotation anticlockwise by ##\theta## why the reflection first if this is ##R_\theta R_x##
 
  • #30
Redwaves said:
I see,
so it's a reflection followed by a rotation anticlockwise by ##\theta## why the reflection first if this is ##R_\theta R_x##
That's the way matrices acting on vectors work: $$Mv = R_\theta (R_xv)$$That means you first reflect ##v##, then rotate the resultant vector.
 
  • #31
I see, I didn't know that.
I guess I understand. I'll practice to make sure I understand properly.
Thank you
 
  • #32
Redwaves said:
From that, the transformation seems to be an rotation of ##\phi - \theta## clockwise and then a reflection over the x axis. Is this correct?
Yes, you are correct. It can be represented by
##
\begin{pmatrix}
\cos(\theta) & \sin(\theta) \\
\sin(\theta) & -\cos(\theta)
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}

\begin{pmatrix}
\cos(-\theta) & -\sin(-\theta) \\
\sin(-\theta) & \cos(-\theta)
\end{pmatrix}
##
That is a rotation of angle ##\theta## in the clockwise direction followed by a reflection across the X-axis.
 

1. What is a transformation matrix?

A transformation matrix is a square matrix that represents a linear transformation in a vector space. It is used to transform a set of coordinates into a new set of coordinates by multiplying the matrix with the original coordinates.

2. How do you find the transformation of a matrix?

The transformation of a matrix can be found by multiplying the original matrix with a transformation matrix. This can be done by performing matrix multiplication, where each element in the resulting matrix is the sum of the products of the corresponding row in the first matrix and the corresponding column in the second matrix.

3. What are the properties of a transformation matrix?

A transformation matrix must be a square matrix, meaning it has the same number of rows and columns. It must also be invertible, meaning it has a unique inverse matrix. Additionally, the transformation matrix must preserve the origin and the distance between points.

4. How do you determine if a matrix is a transformation matrix?

To determine if a matrix is a transformation matrix, you can check if it is a square matrix and if it has a unique inverse. You can also perform a transformation using the matrix and check if it preserves the origin and distances between points.

5. What are some real-world applications of transformation matrices?

Transformation matrices have many applications in fields such as computer graphics, robotics, and physics. They can be used to rotate, scale, and translate objects in 3D space, as well as transform coordinates in GPS systems and map projections. They are also essential in solving systems of linear equations and performing data transformations in statistics.

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