Orthogonal Projection and Reflection: Finding the Image of a Point x = (4,3)

In summary, the student is trying to solve a question from a previous class, but is lost because he doesn't understand what orthogonal projection is.
  • #1
JesseJC
49
0

Homework Statement



|-1/2 -sqrt(3)/2 |
|sqrt(3)/2 -1/2 |


Homework Equations


I don't know

The Attempt at a Solution


Hey everyone, I've been asked to find the "orthogonal projection" on this matrix, this is part B to a question; part A had me use matrix multiplication to find the image of the point x = (4,3) under the reflection of 120 degrees with the positive x-axis. The above matrix was what I came up with, before multipying by x to get some irrational values, of course. But I haven't a clue how to perform orthogonal projection, if anyone could help I'd appreciate it.

Serenity now :0)
 
Physics news on Phys.org
  • #2
The only thing that comes to my mind is the operator ## M M^\dagger ## which is equal to ## MM^T## in your case because all the entries are real.
 
  • Like
Likes JesseJC
  • #3
Shyan said:
The only thing that comes to my mind is the operator ## M M^\dagger ## which is equal to ## MM^T## in your case because all the entries are real.
thanks m8
 
  • #4
What does "orthogonal projection on a matrix" even mean?
 
  • #5
micromass said:
What does "orthogonal projection on a matrix" even mean?
Yeah...that's the question. I don't know why ## |M\rangle\langle M| ## came to my mind.
Now that I think it, it seems to me its the orthogonal projection onto the subspace spanned by the matrix's eigenvectors. If that's the case, at first the eigenvectors should be found. Calling them u and v, the projector is ## uu^\dagger+vv^\dagger ##.
 
  • #6
micromass said:
What does "orthogonal projection on a matrix" even mean?
I don't know, hence the question.
 
  • #7
Shyan said:
Yeah...that's the question. I don't know why ## |M\rangle\langle M| ## came to my mind.
Now that I think it, it seems to me its the orthogonal projection onto the subspace spanned by the matrix's eigenvectors. If that's the case, at first the eigenvectors should be found. Calling them u and v, the projector is ## uu^\dagger+vv^\dagger ##.
We went over it today in lecture, AA^T is all I needed.

Math texts and mathematicians have an incredible way of overcomplicating simple concepts.
 
  • #8
You don't project onto matrices. You project onto subspaces. If the problem doesn't specify which one, you have to find out from the person who asked you to do this. If it's a problem in a book, there should be a definition of the terminology somewhere in the book.

I also don't know what you mean by "reflection of 120 degrees with the positive x-axis". Are you talking about a reflection through the line you get if you rotate the positive x-axis 120 degrees counterclockwise?

You should post the exact problem statement.
 
Last edited:

FAQ: Orthogonal Projection and Reflection: Finding the Image of a Point x = (4,3)

1. What is orthogonal projection?

Orthogonal projection is a mathematical concept used in linear algebra and geometry. It is the process of projecting a vector onto a subspace in a way that preserves the angle between the vector and the subspace. This results in a new vector that is perpendicular (orthogonal) to the subspace.

2. What is the purpose of orthogonal projection?

The purpose of orthogonal projection is to simplify complex vector spaces and make them easier to work with. It also allows for the analysis of a vector in terms of its components, making it useful in solving problems related to linear transformations.

3. How is orthogonal projection calculated?

Orthogonal projection is calculated by finding the dot product between the vector and the basis vectors of the subspace. This dot product is then divided by the norm (length) of the basis vector squared, and multiplied by the basis vector. This process is repeated for each basis vector and the resulting vectors are added together to obtain the orthogonal projection.

4. What is the difference between orthogonal projection and projection?

The main difference between orthogonal projection and projection is the preservation of angles. In regular projection, the angle between the vector and the subspace may change, while in orthogonal projection, the angle remains the same. Additionally, orthogonal projection is only possible when working with vector spaces that have an inner product defined, while regular projection can be used in any vector space.

5. Where is orthogonal projection used in science?

Orthogonal projection has many applications in science, particularly in fields such as physics, engineering, and computer science. It is used in 3D graphics and computer vision for perspective correction, in signal processing for noise reduction, and in physics for analyzing forces and motion. It is also used in data analysis and machine learning for dimensionality reduction and feature extraction.

Similar threads

Replies
1
Views
1K
Replies
11
Views
3K
Replies
1
Views
1K
Replies
6
Views
3K
Replies
2
Views
1K
Replies
46
Views
8K
Replies
3
Views
2K
Back
Top