# Orthogonal projection

1. Jul 24, 2015

### JesseJC

1. The problem statement, all variables and given/known data

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

2. Relevant equations
I don't know

3. 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)

2. Jul 24, 2015

### 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.

3. Jul 24, 2015

### JesseJC

thanks m8

4. Jul 24, 2015

### micromass

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

5. Jul 24, 2015

### ShayanJ

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. Jul 25, 2015

### JesseJC

I don't know, hence the question.

7. Jul 25, 2015

### JesseJC

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. Jul 25, 2015

### Fredrik

Staff Emeritus
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: Jul 25, 2015