1. Jul 17, 2012

Jimmy84

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

A linear operator given (a matrix). That could be an orthogonal protection (that goes through the origin) or a symmetry with respect to a plane (that goes through the origin).

1-Get the eigenvalues of linear operator
2-Get the eigenspace associated with each eigenvalue.

3-Based on the previous calculations determine if the Operator is a symmetry or an orthogonal protection.

4-Describe an ortogonal base of the given plane, and complete it with a base of R^2
The matrix with respect to the calculated base must have the form of the orthogonal projection or of the symmetric matrix

100 or 100
010 010
00-1 000
2. Relevant equations

3. The attempt at a solution

I got the eigenvalues 1 and 0 therefore I'm assuming the operator is an orthogonal projection.

I got the eigenvectors

How can I start to do 4?

Im thinking about using gram schidt to get 3 ortogonal vectors and then to use them as a base .

Thanks a lot for any help, I appreciate it.

2. Jul 17, 2012

ehild

What is the given operator?

ehild

3. Jul 17, 2012

Jimmy84

It is the matrix
2 1 -1
-1 0 1
1 1 0

4. Jul 17, 2012

HallsofIvy

Okay, so this is projection onto a line? What is that line? What is a vector in the direction of that line?

5. Jul 17, 2012

Jimmy84

It seems to be is an orthogonal projection on a plane that goes through the origin.

For now Im confused, I was thinking about using the set of eigenbasis or eigenvectors of the linear operator B . and to use gram schmidt to get 3 orthogonal bases w1, w2 and w3

Im considering to evaluate w1 w2 and w3 with respect to the basis B to get the projection of the plane with respect to B.

6. Jul 17, 2012

HallsofIvy

Once again, what are the eigenvectors corresponding to each eigenvalue?

7. Jul 17, 2012

Jimmy84

the eigenvectors associated with the eigenvalue 1 are -1,1,0 and 1 0 1 .
the ones associated with the eigenvalue 0 are 1 -1 1

8. Jul 18, 2012

ehild

Any linear combination of the eigenvectors belonging to 1 is also an eigenvector to λ=1. Find a combination of a=(1,0,1) and b=(-1,1,0) c=a+kb so the dot product a˙c=0 and choose a and c as orthogonal base in the plane.

ehild