How to Determine if a Linear Operator is a Symmetry or an Orthogonal Projection?

[Jimmy84]

Homework Statement

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

Homework Equations

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.

 

[ehild]

What is the given operator? ehild

 

[Jimmy84]

What is the given operator? ehild

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

 

[HallsofIvy]

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

 

[Jimmy84]

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

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

For now I am 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.

 

[HallsofIvy]

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

 

[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

 

[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