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

[JesseJC]

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)

 

[ShayanJ]

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.

 

[JesseJC]

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

 

[micromass]

What does "orthogonal projection on a matrix" even mean?

 

[ShayanJ]

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 $uu^\dagger+vv^\dagger$.

 

[JesseJC]

What does "orthogonal projection on a matrix" even mean?

I don't know, hence the question.

 

[JesseJC]

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.

 

[Fredrik]

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.