# Eigenvectors, eigenvalues of a transformation

Suppose a linear transformation is defined by [see the attachement for the equation] where x is a vector in the euclidean plane (R^2). Answer the following:

a) What are the eigenvalues and eigenvectors of this transformation?

b) Show that the image of x can be represented as a sum of the eigenvectors.

c) What does this transformation represent?

https://brainmass.com/math/linear-algebra/eigenvectors-eigenvalues-transformation-469484

## SOLUTION This solution is **FREE** courtesy of BrainMass!

From the condition, we have , where .

(a) We set

So the transform has two eigenvalues 1 and -1.

When the eigenvalue is , the eigenvector is

When the eigenvalue is , the eigenvector is

(b) Consider , then we have

Then we get ,

So the image of is a linear combination of two eigenvectors.

(c) This transform maps to , maps to . Let , then the transform is actually a rotation of , followed by a reflection of the y-axis.

https://brainmass.com/math/linear-algebra/eigenvectors-eigenvalues-transformation-469484