SOLUTION: Suppose T : R2 → R2 is the transformation that reflects vectors across the line y = x. Find the eigenvalues and eigenvectors for this transformation.

This transformation leaves UNCHEANGABLE the vectors  (x,y)  in the plane with x=y

that lie on the line  y=x.  

Hence, these vectors  (x,x)  are eigenvectors with the eigenvalue of 1.



This transformation CHANGEs the vectors  (x,y)  in the plane with x=-y

that lie on the line  y=-x,  orthogonal to the line  x=y.  

The transformation transforms each such a vector into the opposite one.

Hence, these vectors  (x,-x) are eigenvectors with the eigenvalue of -1.